keywords: MSC 2020: 60G15; 60H10; 60H15; 60H35; 65C30
Introduction and main results
Weighted sub-fractional Brownian motion is a Gaussian process whose covariance combines a temporal weight with a memory kernel. Its current value generally does not determine the conditional law of future increments. We record the contribution of past Brownian noise to future values in a forward curve, which provides a Markovian state for the driver. Discretizing the same covariance representation gives finitely many Ornstein–Uhlenbeck factors. We study the resulting approximations of both the driver and the scalar equation it drives.
Fix 0 < b < 2 0 < b < 2 0 < b < 2 f ∈ L l o c 1 ( R + ) f \in L_{\mathrm{loc}}^{1}(\mathbb{R}_{+}) f ∈ L loc 1 ( R + ) ζ f , b \zeta_{f,b} ζ f , b
R f , b ( s , t ) = ∫ 0 s ∧ t f ( u ) Q b ( s − u , t − u ) d u , s , t ≥ 0 , (1) R_{f,b}(s,t) = \int_{0}^{s \wedge t} f(u)Q_b(s-u,t-u)\,\mathrm{d}u,\qquad s,t \ge0, \tag*{(1)} R f , b ( s , t ) = ∫ 0 s ∧ t f ( u ) Q b ( s − u , t − u ) d u , s , t ≥ 0 , (1) where
Q b ( x , y ) = { x b + y b − ( x + y ) b 1 − b , b ≠ 1 , ( x + y ) log ( x + y ) − x log x − y log y , b = 1 , x , y ≥ 0 , (2) Q_b(x,y) =
\begin{cases}
\dfrac{x^b+y^b-(x+y)^b}{1-b}, & b \ne1,\\
(x+y)\log(x+y)-x\log x-y\log y, & b=1,
\end{cases}
\qquad x,y \ge0, \tag*{(2)} Q b ( x , y ) = ⎩ ⎨ ⎧ 1 − b x b + y b − ( x + y ) b , ( x + y ) log ( x + y ) − x log x − y log y , b = 1 , b = 1 , x , y ≥ 0 , (2) with 0 log 0 : = 0 0\log0 := 0 0 log 0 := 0 b ≠ 1 b \ne1 b = 1 [18 Section 2.1, Eq. (4)] . The power-weight family originates in occupation-time fluctuation limits of age-dependent branching systems [16 Theorem 2.1(i), Eq. (2.1)] . The logarithmic case b = 1 b = 1 b = 1 [16 Theorem 2.1(ii), p. 596] . Besides its origin in occupation-time fluctuation limits, this Gaussian family has been used in animal movement modeling. Ramírez-González, Murillo-Salas, and Sun [20 ] fitted its logarithmic member to bat trajectories using longitude and latitude observations. The same family is included among the Gaussian models considered in the multidimensional telemetry analysis of [19 ] . Inference for stochastic differential equations driven by weighted sub-fractional Brownian motion is studied in [17 ] . The Euler approximation is used to construct a trajectory likelihood that retains the dependence between the Gaussian driving increments. Neural-network and radial-basis representations are used to estimate the drift and diffusion coefficients together with the normalized temporal weight and covariance exponent.
The regularity of the boundary process changes at the critical value b = 1 b = 1 b = 1 f ∈ L l o c ∞ ( R + ) f \in L_{\mathrm{loc}}^{\infty}(\mathbb{R}_{+}) f ∈ L loc ∞ ( R + ) 0 < b < 1 0 < b < 1 0 < b < 1 ζ f , b \zeta_{f,b} ζ f , b
0 < η < 1 + b 2 . 0 < \eta< \frac{1+b}{2}. 0 < η < 2 1 + b . At b = 1 b = 1 b = 1
H b δ : = { ( 1 + b ) / 2 , 0 < b < 1 , 1 , b = 1 , H_b^\delta:= \begin{cases}
(1+b)/2, & 0<b<1,\\
1, & b=1,
\end{cases} H b δ := { ( 1 + b ) /2 , 1 , 0 < b < 1 , b = 1 , where the superscript indicates that the endpoint is not asserted. Hence the nonlinear equation is a Young equation for 0 < b ≤ 1 0<b\leq1 0 < b ≤ 1 1 < b < 2 1<b<2 1 < b < 2
Relation to the literature.
Laplace representations of fractional Gaussian processes lead to infinite-dimensional Markovian states and finite-factor approximations; see [7 , 8 ] . Affine representations of fractional processes are developed in [14 ] , and strong approximation rates in [13 ] . For stochastic Volterra equations, boundary representations through first-order SPDEs are studied in [5 ] ; weighted Sobolev realizations and polynomially weighted shift spaces are treated in [6 , 15 ] . Related results on affine Volterra processes and their Markovian lifts appear in [1 , 10 , 12 ] .
The nonlinear equations in [6 , 15 ] are formulated as stochastic Volterra equations with Brownian Itô integrals. In our construction, the Gaussian forward curve is also an Itô stochastic evolution, while its boundary drives the scalar equation in the Young or Lebesgue–Stieltjes sense. The main issue for the nonlinear equation is to restart it from an arbitrary curve in the state space: such a curve need not have the Hölder regularity of the Gaussian process. The total-variation bound below provides the required control. For the finite-factor approximations, we use the same Gaussian noise and estimate the driver error in the path norms used by the solution map. This gives convergence under the well-posedness assumptions; quantitative bounds require the additional hypotheses specified below.
The state space and the Hilbert–Schmidt condition.
For
μ b ( d λ ) : = 1 Γ ( 2 − b ) λ 1 − b d λ , λ > 0 , (3) \mu_b(\mathrm{d}\lambda) := \frac{1}{\Gamma(2-b)}\lambda^{1-b}\,\mathrm{d}\lambda,\qquad\lambda>0, \tag*{(3)} μ b ( d λ ) := Γ ( 2 − b ) 1 λ 1 − b d λ , λ > 0 , (3) and
ϕ r ( λ ) : = b 1 − e − λ r λ , r ≥ 0 , (4) \phi_r(\lambda) := \sqrt{b}\frac{1-e^{-\lambda r}}{\lambda},\qquad r\geq0, \tag*{(4)} ϕ r ( λ ) := b λ 1 − e − λ r , r ≥ 0 , (4) let U b : = L 2 ( ( 0 , ∞ ) , μ b ) U_b := L^2((0,\infty),\mu_b) U b := L 2 (( 0 , ∞ ) , μ b ) λ \lambda λ Q b ( x , y ) = ⟨ ϕ x , ϕ y ⟩ U b Q_b(x,y)=\langle\phi_x,\phi_y\rangle_{U_b} Q b ( x , y ) = ⟨ ϕ x , ϕ y ⟩ U b
For 0 < γ < 1 0<\gamma<1 0 < γ < 1
w γ ( r ) : = r γ ( 1 + r ) 2 , r > 0 , (5) w_\gamma(r) := \frac{r^\gamma}{(1+r)^2},\qquad r>0, \tag*{(5)} w γ ( r ) := ( 1 + r ) 2 r γ , r > 0 , (5) and the Hilbert space
E γ : = { θ ∈ A C l o c ( R + ) : ∥ θ ∥ γ 2 : = ∣ θ ( 0 ) ∣ 2 + ∫ 0 ∞ r γ ( 1 + r ) 2 ∣ θ ′ ( r ) ∣ 2 d r < ∞ } . (6) \mathcal{E}_{\gamma}:=\left\{\theta\in AC_{\mathrm{loc}}(\mathbb{R}_{+}):\|\theta\|_{\gamma}^{2}:=|\theta(0)|^{2}+\int_{0}^{\infty}\frac{r^{\gamma}}{(1+r)^{2}}|\theta'(r)|^{2}\,\mathrm{d}r<\infty\right\}. \tag*{(6)} E γ := { θ ∈ A C loc ( R + ) : ∥ θ ∥ γ 2 := ∣ θ ( 0 ) ∣ 2 + ∫ 0 ∞ ( 1 + r ) 2 r γ ∣ θ ′ ( r ) ∣ 2 d r < ∞ } . (6) The value assigned to w γ w_{\gamma} w γ r = 0 r=0 r = 0 r γ r^{\gamma} r γ ( 1 + r ) − 2 (1+r)^{-2} ( 1 + r ) − 2
( G b h ) ( r ) : = ⟨ ϕ r , h ⟩ U b , h ∈ U b , r ≥ 0. (7) (G_bh)(r):=\langle\phi_r,h\rangle_{\mathcal{U}_b},\qquad h\in\mathcal{U}_b,\ r\ge0. \tag*{(7)} ( G b h ) ( r ) := ⟨ ϕ r , h ⟩ U b , h ∈ U b , r ≥ 0. (7) This rule defines a Hilbert–Schmidt operator from U b \mathcal{U}_b U b E γ \mathcal{E}_{\gamma} E γ
b + γ > 1. (8) b+\gamma>1. \tag*{(8)} b + γ > 1. (8) ∥ G b ∥ H S ( U b , E γ ) 2 = b 2 b − 2 Γ ( b + γ − 1 ) Γ ( 3 − b − γ ) . \|G_b\|_{\mathrm{HS}(\mathcal{U}_b,\mathcal{E}_{\gamma})}^{2}=b2^{b-2}\Gamma(b+\gamma-1)\Gamma(3-b-\gamma). ∥ G b ∥ HS ( U b , E γ ) 2 = b 2 b − 2 Γ ( b + γ − 1 ) Γ ( 3 − b − γ ) . The parameters of the lifted equation satisfy
0 < γ < 1 , b + γ > 1. (9) 0<\gamma<1,\qquad b+\gamma>1. \tag*{(9)} 0 < γ < 1 , b + γ > 1. (9) Thus 1 − b < γ < 1 1-b<\gamma<1 1 − b < γ < 1 0 < b < 1 0<b<1 0 < b < 1 0 < γ < 1 0<\gamma<1 0 < γ < 1 1 ≤ b < 2 1\le b<2 1 ≤ b < 2 γ = 0 \gamma=0 γ = 0 1 < b < 2 1<b<2 1 < b < 2 γ \gamma γ
Write ( S t θ ) ( r ) = θ ( t + r ) (S_t\theta)(r)=\theta(t+r) ( S t θ ) ( r ) = θ ( t + r ) A A A ( S t ) t ≥ 0 (S_t)_{t\ge0} ( S t ) t ≥ 0 E γ \mathcal{E}_{\gamma} E γ δ 0 θ = θ ( 0 ) \delta_0\theta=\theta(0) δ 0 θ = θ ( 0 )
d Θ t = A Θ t d t + f ( t ) G b d W t , Θ 0 = 0 , \mathrm{d}\Theta_t=A\Theta_t\,\mathrm{d}t+\sqrt{f(t)}G_b\,\mathrm{d}W_t,\qquad\Theta_0=0, d Θ t = A Θ t d t + f ( t ) G b d W t , Θ 0 = 0 , where W W W U b \mathcal{U}_b U b Θ t ( 0 ) = ζ t , f , b \Theta_t(0)=\zeta_{t,f,b} Θ t ( 0 ) = ζ t , f , b
Θ t ( r ) = E [ ζ t + r , f , b ∣ F t ] . \Theta_t(r)=\mathbb{E}[\zeta_{t+r,f,b}\mid\mathcal{F}_t]. Θ t ( r ) = E [ ζ t + r , f , b ∣ F t ] . Here F t \mathcal{F}_t F t
The nonlinear equation.
Fix T > 0 T>0 T > 0
X t = x 0 + ∫ 0 t a ( X s , s ) d s + ∫ 0 t σ ( X s , s ) d ζ s , f , b , 0 ≤ t ≤ T , (10) X_t=x_0+\int_0^t a(X_s,s)\,\mathrm{d}s+\int_0^t \sigma(X_s,s)\,\mathrm{d}\zeta_{s,f,b},\qquad0\le t\le T, \tag*{(10)} X t = x 0 + ∫ 0 t a ( X s , s ) d s + ∫ 0 t σ ( X s , s ) d ζ s , f , b , 0 ≤ t ≤ T , (10) where x 0 ∈ R x_0\in\mathbb{R} x 0 ∈ R a , σ : R × R + → R a,\sigma:\mathbb{R}\times\mathbb{R}_{+}\to\mathbb{R} a , σ : R × R + → R
Assume additionally that f ∈ L l o c ∞ ( R + ) f \in L^\infty_{\mathrm{loc}}(\mathbb{R}_+) f ∈ L loc ∞ ( R + ) Assumption 3.5 , Theorem 3.10 gives the Markov property for ( X , Θ ) (X,\Theta) ( X , Θ ) C b C_b C b α \alpha α α > ( 1 − b ) / ( 1 + b ) \alpha> (1-b)/(1+b) α > ( 1 − b ) / ( 1 + b ) b < 1 b < 1 b < 1 α > 0 \alpha> 0 α > 0 b = 1 b = 1 b = 1 b > 1 b > 1 b > 1 Table 1 summarizes the parameter ranges for this Hilbert–Schmidt realization and the corresponding pathwise integrals.
Table 1. State exponents and pathwise interpretations used in the lifted theory.
Approximation and coefficient regularity.
Approximating the Laplace features by step functions in the spectral variable produces independent Ornstein–Uhlenbeck factors. The finite driver is the time integral of their weighted sum; the Markovian state for the nonlinear approximation consists of the scalar solution and these factors. Write M M M [ 0 , Λ ] [0,\Lambda] [ 0 , Λ ] δ λ ≤ C m e s h Λ / M \delta_\lambda\le C_{\mathrm{mesh}}\Lambda/M δ λ ≤ C mesh Λ/ M C m e s h C_{\mathrm{mesh}} C mesh M M M Λ \Lambda Λ [ 0 , Λ ] [0,\Lambda] [ 0 , Λ ] Theorem 4.1 gives the strong solution rate M − ( b − 1 ) / 3 M^{-(b-1)/3} M − ( b − 1 ) /3 1 < b < 2 1 < b < 2 1 < b < 2 Λ = M 2 / 3 \Lambda= M^{2/3} Λ = M 2/3 Theorem 4.3 gives the uniform driver rate M − b / ( 4 p ) log ( 2 + M ) M^{-b/(4p)}\sqrt{\log(2+M)} M − b / ( 4 p ) log ( 2 + M ) Λ = M 1 / 2 \Lambda= M^{1/2} Λ = M 1/2
( E sup t ≤ T ∣ X t − X t M , Λ ∣ p ) 1 / p ≤ C ( M − b / ( 4 p ) log ( 2 + M ) ) 1 − η / η ′ , Λ = M 1 / 2 . \left(\mathbb{E}\sup_{t\le T}\left|X_t-X_t^{M,\Lambda}\right|^p\right)^{1/p}
\le C\left(M^{-b/(4p)}\sqrt{\log(2+M)}\right)^{1-\eta/\eta'},\qquad\Lambda=M^{1/2}. ( E t ≤ T sup X t − X t M , Λ p ) 1/ p ≤ C ( M − b / ( 4 p ) log ( 2 + M ) ) 1 − η / η ′ , Λ = M 1/2 . for p ≥ 2 p \ge2 p ≥ 2 1 / 2 < η < η ′ < H b ∘ 1/2 < \eta< \eta' < H_b^\circ 1/2 < η < η ′ < H b ∘ Proposition 4.8 establishes qualitative convergence in every finite L p L^p L p 0 < b ≤ 1 0 < b \le1 0 < b ≤ 1
Assumption 4.11 ⟹ Assumption 4.4 ⟹ Assumption 3.5 , \text{{Assumption 4.11}}\ \Longrightarrow\ \text{{Assumption 4.4}}\ \Longrightarrow\ \text{{Assumption 3.5}}, Assumption 4.11 ⟹ Assumption 4.4 ⟹ Assumption 3.5 , where the last implication holds with α = 1 \alpha= 1 α = 1 Assumption 3.5 gives the Markov–Feller property and qualitative convergence. Assumption 4.4 also bounds the diffusion and imposes joint spatial and temporal Lipschitz continuity on its derivative; these bounds are used for quantitative stability and the spectral solution rate. Assumption 4.11 adds a bounded drift, uniformly Lipschitz in time, for the Euler estimate and the fully discrete bound. For 1 < b < 2 1 < b < 2 1 < b < 2
Theorem 4.13 bounds separately the spectral error, the factors’ time-discretization error, and the scalar Euler error. The dependence on the spectral cutoff remains explicit, since increasing that cutoff requires finer time resolution. Corollary 4.14 gives sufficient conditions on the spectral and time meshes to preserve the spectral convergence rate.
Organization
Section 2 develops the Laplace factorization, the polynomially weighted state space, and the forward-curve lift. Section 3 proves the path regularity, the boundary-slope criterion, and the Markov–Feller lift of (10) . Section 4 establishes the finite-factor and fully discrete convergence bounds.
Notation and conventions
All stochastic processes are defined on a complete filtered probability space ( Ω , F , ( F t ) t ≥ 0 , P ) (\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge0}, \mathbb{P}) ( Ω , F , ( F t ) t ≥ 0 , P ) E \mathbb{E} E P \mathbb{P} P Γ \Gamma Γ Tr \operatorname{Tr} Tr N : = { 1 , 2 , … } \mathbb{N} := \{1, 2, \ldots\} N := { 1 , 2 , … } R + = [ 0 , ∞ ) \mathbb{R}_{+} = [0, \infty) R + = [ 0 , ∞ ) A C l o c ( R + ) AC_{\mathrm{loc}}(\mathbb{R}_{+}) A C loc ( R + ) R + \mathbb{R}_{+} R + E E E C b ( E ) C_b(E) C b ( E ) E E E U U U E E E L ( U , E ) \mathcal{L}(U,E) L ( U , E ) U U U E E E ∥ T ∥ o p : = sup ∥ u ∥ U ≤ 1 ∥ T u ∥ E \|T\|_{\mathrm{op}} := \sup_{\|u\|_U \le1} \|Tu\|_E ∥ T ∥ op := sup ∥ u ∥ U ≤ 1 ∥ T u ∥ E g : R → R g : \mathbb{R} \to\mathbb{R} g : R → R Lip ( g ) : = sup x ≠ y ∣ g ( x ) − g ( y ) ∣ / ∣ x − y ∣ \operatorname{Lip}(g) := \sup_{x \ne y} |g(x) - g(y)|/|x-y| Lip ( g ) := sup x = y ∣ g ( x ) − g ( y ) ∣/∣ x − y ∣ U U U E E E T ∈ L ( U , E ) T \in\mathcal{L}(U,E) T ∈ L ( U , E ) ( e k ) (e_k) ( e k ) U U U
∑ k = 1 ∞ ∥ T e k ∥ E 2 < ∞ . \sum_{k=1}^{\infty} \|Te_k\|_E^2 < \infty. k = 1 ∑ ∞ ∥ T e k ∥ E 2 < ∞. This condition and the value of the sum are independent of the chosen basis; we denote the corresponding space by L 2 ( U , E ) \mathcal{L}_2(U,E) L 2 ( U , E ) ∥ T ∥ H S ( U , E ) \|T\|_{\mathrm{HS}(U,E)} ∥ T ∥ HS ( U , E ) T ∗ T^* T ∗ T T T T T T D ( T ) D(T) D ( T ) ⟨ ⋅ , ⋅ ⟩ U \langle\cdot, \cdot\rangle_U ⟨ ⋅ , ⋅ ⟩ U ∥ ⋅ ∥ U \|\cdot\|_U ∥ ⋅ ∥ U U U U Z Z Z p ≥ 1 p \ge1 p ≥ 1 ∥ Z ∥ L p ( Ω ) : = ( E ∣ Z ∣ p ) 1 / p \|Z\|_{L^p(\Omega)} := (\mathbb{E}|Z|^p)^{1/p} ∥ Z ∥ L p ( Ω ) := ( E ∣ Z ∣ p ) 1/ p x ∈ C ( [ 0 , T ] ) x \in C([0,T]) x ∈ C ([ 0 , T ]) η ∈ ( 0 , 1 ) \eta\in(0,1) η ∈ ( 0 , 1 )
∥ x ∥ ∞ ; [ 0 , T ] : = sup 0 ≤ t ≤ T ∣ x ( t ) ∣ , [ x ] η ; [ 0 , T ] : = sup 0 ≤ s < t ≤ T ∣ x ( t ) − x ( s ) ∣ ∣ t − s ∣ η . \|x\|_{\infty;[0,T]} := \sup_{0 \le t \le T} |x(t)|, \qquad[x]_{\eta;[0,T]} := \sup_{0 \le s < t \le T} \frac{|x(t)-x(s)|}{|t-s|^\eta}. ∥ x ∥ ∞ ; [ 0 , T ] := 0 ≤ t ≤ T sup ∣ x ( t ) ∣ , [ x ] η ; [ 0 , T ] := 0 ≤ s < t ≤ T sup ∣ t − s ∣ η ∣ x ( t ) − x ( s ) ∣ . We write C η ( [ 0 , T ] ) : = { x ∈ C ( [ 0 , T ] ) : [ x ] η ; [ 0 , T ] < ∞ } C^\eta([0,T]) := \{x \in C([0,T]) : [x]_{\eta;[0,T]} < \infty\} C η ([ 0 , T ]) := { x ∈ C ([ 0 , T ]) : [ x ] η ; [ 0 , T ] < ∞ }
∥ x ∥ C η ( [ 0 , T ] ) : = ∥ x ∥ ∞ ; [ 0 , T ] + [ x ] η ; [ 0 , T ] . \|x\|_{C^\eta([0,T])} := \|x\|_{\infty;[0,T]} + [x]_{\eta;[0,T]}. ∥ x ∥ C η ([ 0 , T ]) := ∥ x ∥ ∞ ; [ 0 , T ] + [ x ] η ; [ 0 , T ] . For x ∈ C ( [ 0 , T ] ) x \in C([0,T]) x ∈ C ([ 0 , T ]) p ≥ 1 p \ge1 p ≥ 1
∥ x ∥ p -var ; [ 0 , T ] : = ( sup Π ∑ i = 0 m − 1 ∣ x ( t i + 1 ) − x ( t i ) ∣ p ) 1 / p . \lVert x\rVert_{p\text{-var};[0,T]} := \left(\sup_{\Pi}\sum_{i=0}^{m-1}\lvert x(t_{i+1})-x(t_i)\rvert^p\right)^{1/p}. ∥ x ∥ p -var ; [ 0 , T ] := ( Π sup i = 0 ∑ m − 1 ∣ x ( t i + 1 ) − x ( t i ) ∣ p ) 1/ p . where the supremum is over all finite partitions Π = { 0 = t 0 < ⋯ < t m = T } \Pi= \{0=t_0<\cdots<t_m=T\} Π = { 0 = t 0 < ⋯ < t m = T }
TV [ 0 , T ] ( x ) : = sup Π ∑ i = 0 m − 1 ∣ x ( t i + 1 ) − x ( t i ) ∣ . \operatorname{TV}_{[0,T]}(x) := \sup_{\Pi}\sum_{i=0}^{m-1}\lvert x(t_{i+1})-x(t_i)\rvert. TV [ 0 , T ] ( x ) := Π sup i = 0 ∑ m − 1 ∣ x ( t i + 1 ) − x ( t i )∣ . For a finite partition π \pi π ∣ π ∣ : = max [ u , v ] ∈ π ( v − u ) \lvert\pi\rvert:= \max_{[u,v]\in\pi}(v-u) ∣ π ∣ := max [ u , v ] ∈ π ( v − u ) f ∈ L ∞ ( 0 , T ) f \in L^\infty(0,T) f ∈ L ∞ ( 0 , T )
∥ f ∥ ∞ , T : = ess sup 0 ≤ t ≤ T ∣ f ( t ) ∣ . \lVert f\rVert_{\infty,T} := \operatorname*{ess\,sup}_{0\le t\le T}\lvert f(t)\rvert. ∥ f ∥ ∞ , T := 0 ≤ t ≤ T ess sup ∣ f ( t )∣ . For processes X X X Y Y Y X = f . d . d . Y X \mathrel{\overset{\mathrm{f.d.d.}}{=}} Y X = f.d.d. Y ( X t 1 , … , X t m ) = d ( Y t 1 , … , Y t m ) (X_{t_1},\ldots,X_{t_m}) \mathrel{\overset{\mathrm{d}}{=}} (Y_{t_1},\ldots,Y_{t_m}) ( X t 1 , … , X t m ) = d ( Y t 1 , … , Y t m ) m ∈ N m \in\mathbb{N} m ∈ N t 1 , … , t m ≥ 0 t_1,\ldots,t_m \ge0 t 1 , … , t m ≥ 0 C C C C C C ( P s , t ) s ≤ t (P_{s,t})_{s\le t} ( P s , t ) s ≤ t E E E P s , t C b ( E ) ⊆ C b ( E ) P_{s,t}C_b(E) \subseteq C_b(E) P s , t C b ( E ) ⊆ C b ( E ) s ≤ t s \le t s ≤ t C b C_b C b C b C_b C b f f f L l o c 1 L^1_{\mathrm{loc}} L loc 1 L l o c ∞ L^\infty_{\mathrm{loc}} L loc ∞
Covariance factorization and the Markovian lift
We first factor the covariance into Laplace features. We then choose a Sobolev space on which right translation is strongly continuous and the feature operator is Hilbert–Schmidt. The resulting stochastic convolution has the Gaussian driver as its boundary value and the Markov property as a curve-valued process.
The covariance features
For 0 < b < 2 0 < b < 2 0 < b < 2 μ b \mu_b μ b ϕ r \phi_r ϕ r
κ b : = { 2 − 2 b 1 − b , b ≠ 1 , 2 log 2 , b = 1. \kappa_b :=
\begin{cases}
\dfrac{2-2^b}{1-b}, & b \ne1,\\
2\log2, & b=1.
\end{cases} κ b := ⎩ ⎨ ⎧ 1 − b 2 − 2 b , 2 log 2 , b = 1 , b = 1. For b ≠ 1 b \ne1 b = 1 , {xrefext}Eq. (6)]. We formulate it in the spectral Hilbert space used below, with the same normalization at b = 1 b=1 b = 1
Proposition 2.1. For every 0 < b < 2 0<b<2 0 < b < 2 and x , y ≥ 0 x,y\ge0 x , y ≥ 0
Q b ( x , y ) = ⟨ ϕ x , ϕ y ⟩ U b = ∫ 0 ∞ ϕ x ( λ ) ϕ y ( λ ) μ b ( d λ ) , (11) Q_b(x,y)=\langle\phi_x,\phi_y\rangle_{U_b}=\int_0^\infty\phi_x(\lambda)\phi_y(\lambda)\mu_b(\mathrm{d}\lambda), \tag*{(11)} Q b ( x , y ) = ⟨ ϕ x , ϕ y ⟩ U b = ∫ 0 ∞ ϕ x ( λ ) ϕ y ( λ ) μ b ( d λ ) , (11) where Q b Q_b Q b (2) . In particular,
∥ ϕ r ∥ U b 2 = Q b ( r , r ) = κ b r b . (12) \lVert\phi_r\rVert_{U_b}^2=Q_b(r,r)=\kappa_b r^b. \tag*{(12)} ∥ ϕ r ∥ U b 2 = Q b ( r , r ) = κ b r b . (12) The kernel Q b Q_b Q b x 1 , … , x n > 0 x_1,\ldots,x_n>0 x 1 , … , x n > 0 ( Q b ( x i , x j ) ) i , j = 1 n \bigl(Q_b(x_i,x_j)\bigr)_{i,j=1}^n ( Q b ( x i , x j ) ) i , j = 1 n
Proof. By definition,
⟨ ϕ x , ϕ y ⟩ U b = b Γ ( 2 − b ) ∫ 0 ∞ ( 1 − e − λ x ) ( 1 − e − λ y ) λ − 1 − b d λ . \langle\phi_x,\phi_y\rangle_{U_b}=\frac{b}{\Gamma(2-b)}\int_0^\infty(1-e^{-\lambda x})(1-e^{-\lambda y})\lambda^{-1-b}\,\mathrm{d}\lambda. ⟨ ϕ x , ϕ y ⟩ U b = Γ ( 2 − b ) b ∫ 0 ∞ ( 1 − e − λ x ) ( 1 − e − λ y ) λ − 1 − b d λ . For x , y > 0 x,y>0 x , y > 0 λ 1 − b \lambda^{1-b} λ 1 − b λ − 1 − b \lambda^{-1-b} λ − 1 − b 0 < b < 2 0<b<2 0 < b < 2
( 1 − e − λ x ) ( 1 − e − λ y ) = λ 2 ∫ 0 x ∫ 0 y e − λ ( v + w ) d w d v (1-e^{-\lambda x})(1-e^{-\lambda y})=\lambda^2\int_0^x\int_0^y e^{-\lambda(v+w)}\,\mathrm{d}w\,\mathrm{d}v ( 1 − e − λ x ) ( 1 − e − λ y ) = λ 2 ∫ 0 x ∫ 0 y e − λ ( v + w ) d w d v and Tonelli’s theorem, we obtain
⟨ ϕ x , ϕ y ⟩ U b = ∫ 0 x ∫ 0 y b Γ ( 2 − b ) ∫ 0 ∞ e − λ ( v + w ) λ 1 − b d λ d w d v = ∫ 0 x ∫ 0 y b ( v + w ) b − 2 d w d v . \langle\phi_x,\phi_y\rangle_{U_b}=\int_0^x\int_0^y\frac{b}{\Gamma(2-b)}\int_0^\infty e^{-\lambda(v+w)}\lambda^{1-b}\,\mathrm{d}\lambda\,\mathrm{d}w\,\mathrm{d}v
=\int_0^x\int_0^y b(v+w)^{b-2}\,\mathrm{d}w\,\mathrm{d}v. ⟨ ϕ x , ϕ y ⟩ U b = ∫ 0 x ∫ 0 y Γ ( 2 − b ) b ∫ 0 ∞ e − λ ( v + w ) λ 1 − b d λ d w d v = ∫ 0 x ∫ 0 y b ( v + w ) b − 2 d w d v . The singularity at ( v , w ) = ( 0 , 0 ) (v,w)=(0,0) ( v , w ) = ( 0 , 0 ) v + w = s v+w=s v + w = s s s s ∫ 0 x + y s b − 1 d s \int_0^{x+y}s^{b-1}\,\mathrm{d}s ∫ 0 x + y s b − 1 d s b ≠ 1 b\ne1 b = 1
∫ 0 x ∫ 0 y b ( v + w ) b − 2 d w d v = ( x + y ) b − x b − y b b − 1 = Q b ( x , y ) . \int_0^x\int_0^y b(v+w)^{b-2}\,\mathrm{d}w\,\mathrm{d}v=\frac{(x+y)^b-x^b-y^b}{b-1}=Q_b(x,y). ∫ 0 x ∫ 0 y b ( v + w ) b − 2 d w d v = b − 1 ( x + y ) b − x b − y b = Q b ( x , y ) . For b = 1 b=1 b = 1
∫ 0 x log v + y v d v = ( x + y ) log ( x + y ) − x log x − y log y = Q 1 ( x , y ) . \int_0^x\log\frac{v+y}{v}\,\mathrm{d}v=(x+y)\log(x+y)-x\log x-y\log y=Q_1(x,y). ∫ 0 x log v v + y d v = ( x + y ) log ( x + y ) − x log x − y log y = Q 1 ( x , y ) . If either coordinate is zero, both sides vanish. This proves (11) in all cases, and setting x = y = r x=y=r x = y = r (12) .
To prove infinite rank, fix distinct positive numbers x 1 , … , x n x_1,\ldots,x_n x 1 , … , x n ∑ j = 1 n c j ϕ x j = 0 \sum_{j=1}^n c_j\phi_{x_j}=0 ∑ j = 1 n c j ϕ x j = 0 U b U_b U b λ > 0 \lambda>0 λ > 0 μ b \mu_b μ b λ > 0 \lambda>0 λ > 0 λ / b \lambda/\sqrt{b} λ / b
∑ j = 1 n c j ( 1 − e − λ x j ) = 0 , λ > 0. \sum_{j=1}^n c_j(1-e^{-\lambda x_j})=0,\qquad\lambda>0. j = 1 ∑ n c j ( 1 − e − λ x j ) = 0 , λ > 0. Letting λ → ∞ \lambda\to\infty λ → ∞ ∑ j c j = 0 \sum_j c_j=0 ∑ j c j = 0 ∑ j c j e − λ x j = 0 \sum_j c_j e^{-\lambda x_j}=0 ∑ j c j e − λ x j = 0 λ > 0 \lambda>0 λ > 0 x 1 < ⋯ < x n x_1<\cdots<x_n x 1 < ⋯ < x n e λ x 1 e^{\lambda x_1} e λ x 1 λ → ∞ \lambda\to\infty λ → ∞ c 1 = 0 c_1=0 c 1 = 0 c 1 = ⋯ = c n = 0 c_1=\cdots=c_n=0 c 1 = ⋯ = c n = 0 c = ( c 1 , … , c n ) c=(c_1,\ldots,c_n) c = ( c 1 , … , c n )
∑ i , j = 1 n c i c j Q b ( x i , x j ) = ∥ ∑ j = 1 n c j ϕ x j ∥ U b 2 > 0 , \sum_{i,j=1}^n c_i c_j Q_b(x_i,x_j)=\left\lVert\sum_{j=1}^n c_j\phi_{x_j}\right\rVert_{U_b}^2>0, i , j = 1 ∑ n c i c j Q b ( x i , x j ) = j = 1 ∑ n c j ϕ x j U b 2 > 0 , so the corresponding Gram matrix is strictly positive definite. Since every nontrivial interval contains arbitrarily large finite collections of distinct points, the Gram kernel has infinite rank there. □
Corollary 2.2 Fix 0 < b < 2 0 < b < 2 0 < b < 2 N ∈ N N \in\mathbb{N} N ∈ N k b , 1 , … , k b , N : R + → R k_{b,1}, \ldots, k_{b,N} : \mathbb{R}_{+} \to\mathbb{R} k b , 1 , … , k b , N : R + → R
Q b ( x , y ) = ∑ j = 1 N k b , j ( x ) k b , j ( y ) , x , y ≥ 0. (13) Q_b(x,y) = \sum_{j=1}^{N} k_{b,j}(x)k_{b,j}(y), \qquad x,y \ge0. \tag*{(13)} Q b ( x , y ) = j = 1 ∑ N k b , j ( x ) k b , j ( y ) , x , y ≥ 0. (13) Consequently, there is no representation valid for every non-negative, locally bounded weight f f f
ζ t , f , b = f . d . d . ∑ j = 1 N ∫ 0 t f ( u ) k b , j ( t − u ) d B u j . (14) \zeta_{t,f,b} \mathrel{\stackrel{\mathrm{f.d.d.}}{=}} \sum_{j=1}^{N}\int_0^t \sqrt{f(u)}k_{b,j}(t-u)\,\mathrm{d}B_u^j. \tag*{(14)} ζ t , f , b = f.d.d. j = 1 ∑ N ∫ 0 t f ( u ) k b , j ( t − u ) d B u j . (14) where B 1 , … , B N B^1,\ldots,B^N B 1 , … , B N k b , j k_{b,j} k b , j L l o c 2 ( R + ) L^2_{\mathrm{loc}}(\mathbb{R}_{+}) L loc 2 ( R + ) f f f
Proof. If (13) held, every Gram matrix of Q b Q_b Q b N N N N + 1 N+1 N + 1 ( N + 1 ) × ( N + 1 ) (N+1) \times(N+1) ( N + 1 ) × ( N + 1 ) f f f s , t > 0 s,t>0 s , t > 0
∫ 0 s ∧ t f ( u ) D s , t ( u ) d u = 0 , (15) \int_0^{s\wedge t} f(u)D_{s,t}(u)\,\mathrm{d}u = 0, \tag*{(15)} ∫ 0 s ∧ t f ( u ) D s , t ( u ) d u = 0 , (15) where
D s , t ( u ) : = Q b ( s − u , t − u ) − ∑ j = 1 N k b , j ( s − u ) k b , j ( t − u ) . D_{s,t}(u) := Q_b(s-u,t-u) - \sum_{j=1}^{N} k_{b,j}(s-u)k_{b,j}(t-u). D s , t ( u ) := Q b ( s − u , t − u ) − j = 1 ∑ N k b , j ( s − u ) k b , j ( t − u ) . The local L 2 L^2 L 2 D s , t ∈ L 1 ( 0 , s ∧ t ) D_{s,t} \in L^1(0,s\wedge t) D s , t ∈ L 1 ( 0 , s ∧ t ) ( 0 , s ∧ t ) (0,s\wedge t) ( 0 , s ∧ t ) D s , t = 0 D_{s,t}=0 D s , t = 0
Fix d ≥ 0 d \ge0 d ≥ 0 L > 0 L>0 L > 0 s = L s=L s = L t = L + d t=L+d t = L + d x = L − u x=L-u x = L − u
Q b ( x , x + d ) = ∑ j = 1 N k b , j ( x ) k b , j ( x + d ) for almost every x ∈ ( 0 , L ) . (16) Q_b(x,x+d) = \sum_{j=1}^{N} k_{b,j}(x)k_{b,j}(x+d)\quad\text{for almost every }x\in(0,L). \tag*{(16)} Q b ( x , x + d ) = j = 1 ∑ N k b , j ( x ) k b , j ( x + d ) for almost every x ∈ ( 0 , L ) . (16) Since L L L x > 0 x>0 x > 0 d ≥ 0 d\ge0 d ≥ 0 ( x , d ) (x,d) ( x , d ) ( x , d ) ↦ ( x , x + d ) (x,d)\mapsto(x,x+d) ( x , d ) ↦ ( x , x + d )
Q b ( x , y ) = ∑ j = 1 N k b , j ( x ) k b , j ( y ) for almost every ( x , y ) ∈ ( 0 , ∞ ) 2 . (17) Q_b(x,y) = \sum_{j=1}^{N} k_{b,j}(x)k_{b,j}(y)\quad\text{for almost every }(x,y)\in(0,\infty)^2. \tag*{(17)} Q b ( x , y ) = j = 1 ∑ N k b , j ( x ) k b , j ( y ) for almost every ( x , y ) ∈ ( 0 , ∞ ) 2 . (17) Equation (16) with d = 0 d=0 d = 0
Let I I I ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) N + 1 N+1 N + 1 I I I x 1 , … , x N + 1 ∈ I x_1,\ldots,x_{N+1}\in I x 1 , … , x N + 1 ∈ I
( Q b ( x i , x j ) ) i , j = 1 N + 1 = ( ∑ ℓ = 1 N k b , ℓ ( x i ) k b , ℓ ( x j ) ) i , j = 1 N + 1 . \left(Q_b(x_i,x_j)\right)_{i,j=1}^{N+1}
=
\left(\sum_{\ell=1}^{N} k_{b,\ell}(x_i)k_{b,\ell}(x_j)\right)_{i,j=1}^{N+1}. ( Q b ( x i , x j ) ) i , j = 1 N + 1 = ( ℓ = 1 ∑ N k b , ℓ ( x i ) k b , ℓ ( x j ) ) i , j = 1 N + 1 . The matrix on the right has rank at most N N N
The corollary applies to a fixed finite family of convolution kernels that works for every non-negative locally bounded weight. It does not rule out other finite-dimensional states or kernels chosen for a fixed weight.
Polynomially weighted Sobolev spaces
For 0 ≤ γ < 1 0 \le\gamma< 1 0 ≤ γ < 1 w γ w_\gamma w γ E γ E_\gamma E γ
J γ ( T ) : = ∫ 0 T 1 w γ ( r ) d r = T 1 − γ 1 − γ + 2 T 2 − γ 2 − γ + T 3 − γ 3 − γ < ∞ . (18) J_\gamma(T) := \int_0^T \frac{1}{w_\gamma(r)}\,\mathrm{d}r = \frac{T^{1-\gamma}}{1-\gamma} + \frac{2T^{2-\gamma}}{2-\gamma} + \frac{T^{3-\gamma}}{3-\gamma} < \infty. \tag*{(18)} J γ ( T ) := ∫ 0 T w γ ( r ) 1 d r = 1 − γ T 1 − γ + 2 − γ 2 T 2 − γ + 3 − γ T 3 − γ < ∞. (18) Lemma 2.3 Let 0 ≤ γ < 1 0 \le\gamma< 1 0 ≤ γ < 1 E γ E_\gamma E γ θ ∈ E γ \theta\in E_\gamma θ ∈ E γ T > 0 T > 0 T > 0
TV [ 0 , T ] ( θ ) ≤ J γ ( T ) 1 / 2 ∥ θ ∥ γ . (19) \operatorname{TV}_{[0,T]}(\theta) \le J_\gamma(T)^{1/2}\|\theta\|_\gamma. \tag*{(19)} TV [ 0 , T ] ( θ ) ≤ J γ ( T ) 1/2 ∥ θ ∥ γ . (19) Boundary evaluation δ 0 θ = θ ( 0 ) \delta_0\theta= \theta(0) δ 0 θ = θ ( 0 ) ( S t θ ) ( r ) : = θ ( t + r ) (S_t\theta)(r) := \theta(t+r) ( S t θ ) ( r ) := θ ( t + r ) E γ E_\gamma E γ
A θ = θ ′ , D ( A ) = { θ ∈ E γ : θ ′ ∈ E γ } . (20) A\theta= \theta', \qquad D(A) = \{\theta\in E_\gamma: \theta' \in E_\gamma\}. \tag*{(20)} A θ = θ ′ , D ( A ) = { θ ∈ E γ : θ ′ ∈ E γ } . (20) Here θ ′ ∈ E γ \theta' \in E_\gamma θ ′ ∈ E γ t ≥ 0 t \ge0 t ≥ 0
∥ S t ∥ L ( E γ ) 2 ≤ ( 1 + J γ ( t ) 1 / 2 ) 2 + ( 1 + t ) 2 . (21) \|S_t\|_{\mathcal{L}(E_\gamma)}^2 \le(1+J_\gamma(t)^{1/2})^2 + (1+t)^2. \tag*{(21)} ∥ S t ∥ L ( E γ ) 2 ≤ ( 1 + J γ ( t ) 1/2 ) 2 + ( 1 + t ) 2 . (21) In particular, for each T > 0 T > 0 T > 0 C γ , T < ∞ C_{\gamma,T} < \infty C γ , T < ∞ sup 0 ≤ t ≤ T ∥ S t ∥ L ( E γ ) ≤ C γ , T \sup_{0 \le t \le T}\|S_t\|_{\mathcal{L}(E_\gamma)} \le C_{\gamma,T} sup 0 ≤ t ≤ T ∥ S t ∥ L ( E γ ) ≤ C γ , T
Proof. Set
H γ : = L 2 ( R + , r γ ( 1 + r ) 2 d r ) . H_\gamma:= L^2\left(\mathbb{R}_+,\frac{r^\gamma}{(1+r)^2}\,\mathrm{d}r\right). H γ := L 2 ( R + , ( 1 + r ) 2 r γ d r ) . With the inner product
⟨ θ , ϑ ⟩ γ : = θ ( 0 ) ϑ ( 0 ) + ∫ 0 ∞ w γ ( r ) θ ′ ( r ) ϑ ′ ( r ) d r , \langle\theta,\vartheta\rangle_\gamma:= \theta(0)\vartheta(0) + \int_0^\infty w_\gamma(r)\theta'(r)\vartheta'(r)\,\mathrm{d}r, ⟨ θ , ϑ ⟩ γ := θ ( 0 ) ϑ ( 0 ) + ∫ 0 ∞ w γ ( r ) θ ′ ( r ) ϑ ′ ( r ) d r , the map J θ : = ( θ ( 0 ) , θ ′ ) \mathcal{J}\theta:= (\theta(0),\theta') J θ := ( θ ( 0 ) , θ ′ ) E γ E_\gamma E γ R × H γ \mathbb{R} \times H_\gamma R × H γ ( a , h ) ∈ R × H γ (a,h) \in\mathbb{R} \times H_\gamma ( a , h ) ∈ R × H γ
θ ( r ) : = a + ∫ 0 r h ( q ) d q \theta(r) := a + \int_0^r h(q)\,\mathrm{d}q θ ( r ) := a + ∫ 0 r h ( q ) d q is locally absolutely continuous. Indeed, by Cauchy–Schwarz and (18),
∫ 0 T ∣ h ( r ) ∣ d r ≤ J γ ( T ) 1 / 2 ( ∫ 0 T r γ ( 1 + r ) 2 ∣ h ( r ) ∣ 2 d r ) 1 / 2 < ∞ . \int_0^T |h(r)|\,\mathrm{d}r \le J_\gamma(T)^{1/2}\left(\int_0^T \frac{r^\gamma}{(1+r)^2}|h(r)|^2\,\mathrm{d}r\right)^{1/2} < \infty. ∫ 0 T ∣ h ( r ) ∣ d r ≤ J γ ( T ) 1/2 ( ∫ 0 T ( 1 + r ) 2 r γ ∣ h ( r ) ∣ 2 d r ) 1/2 < ∞. Thus J \mathcal{J} J R × H γ \mathbb{R} \times H_\gamma R × H γ E γ E_\gamma E γ ∫ 0 T ∣ θ ′ ( r ) ∣ d r \int_0^T |\theta'(r)|\,\mathrm{d}r ∫ 0 T ∣ θ ′ ( r ) ∣ d r ∣ δ 0 θ ∣ ≤ ∥ θ ∥ γ |\delta_0\theta| \le\|\theta\|_\gamma ∣ δ 0 θ ∣ ≤ ∥ θ ∥ γ u > t ≥ 0 u > t \ge0 u > t ≥ 0
w γ ( u − t ) w γ ( u ) = ( u − t u ) γ ( 1 + u 1 + u − t ) 2 ≤ ( 1 + t ) 2 , \frac{w_\gamma(u-t)}{w_\gamma(u)} = \left(\frac{u-t}{u}\right)^\gamma\left(\frac{1+u}{1+u-t}\right)^2 \le(1+t)^2, w γ ( u ) w γ ( u − t ) = ( u u − t ) γ ( 1 + u − t 1 + u ) 2 ≤ ( 1 + t ) 2 , where the first factor is at most one and ( 1 + u ) / ( 1 + u − t ) = 1 + t / ( 1 + u − t ) ≤ 1 + t (1+u)/(1+u-t)=1+t/(1+u-t)\le1+t ( 1 + u ) / ( 1 + u − t ) = 1 + t / ( 1 + u − t ) ≤ 1 + t
∫ 0 ∞ w γ ( r ) ∣ ( S t θ ) ′ ( r ) ∣ 2 d r = ∫ t ∞ w γ ( u − t ) ∣ θ ′ ( u ) ∣ 2 d u ≤ ( 1 + t ) 2 ∫ t ∞ w γ ( u ) ∣ θ ′ ( u ) ∣ 2 d u . \int_{0}^{\infty} w_{\gamma}(r)|(S_t\theta)'(r)|^2\,dr=\int_{t}^{\infty}w_{\gamma}(u-t)|\theta'(u)|^2\,du
\le(1+t)^2\int_{t}^{\infty}w_{\gamma}(u)|\theta'(u)|^2\,du. ∫ 0 ∞ w γ ( r ) ∣ ( S t θ ) ′ ( r ) ∣ 2 d r = ∫ t ∞ w γ ( u − t ) ∣ θ ′ ( u ) ∣ 2 d u ≤ ( 1 + t ) 2 ∫ t ∞ w γ ( u ) ∣ θ ′ ( u ) ∣ 2 d u . Moreover,
∣ θ ( t ) ∣ ≤ ∣ θ ( 0 ) ∣ + ∫ 0 t ∣ θ ′ ( r ) ∣ d r ≤ ( 1 + J γ ( t ) 1 / 2 ) ∥ θ ∥ γ . |\theta(t)|\le|\theta(0)|+\int_{0}^{t}|\theta'(r)|\,dr\le(1+J_{\gamma}(t)^{1/2})\|\theta\|_{\gamma}. ∣ θ ( t ) ∣ ≤ ∣ θ ( 0 ) ∣ + ∫ 0 t ∣ θ ′ ( r ) ∣ d r ≤ ( 1 + J γ ( t ) 1/2 ) ∥ θ ∥ γ . Combining the last two estimates gives (21). In particular, every S t S_t S t
To prove strong continuity, let τ t h ( r ) : = h ( t + r ) \tau_t h(r):=h(t+r) τ t h ( r ) := h ( t + r ) H γ H_\gamma H γ h ∈ C c ( R + ) h\in C_c(\mathbb{R}_+) h ∈ C c ( R + ) τ t h → h \tau_t h\to h τ t h → h t t t ∥ τ t h − h ∥ H γ → 0 \|\tau_t h-h\|_{H_\gamma}\to0 ∥ τ t h − h ∥ H γ → 0 C c ( R + ) C_c(\mathbb{R}_+) C c ( R + ) H γ H_\gamma H γ h ∈ H γ h\in H_\gamma h ∈ H γ J J J
J S t J − 1 ( a , h ) = ( a + ∫ 0 t h ( s ) d s , τ t h ) . (22) JS_tJ^{-1}(a,h)=\left(a+\int_{0}^{t}h(s)\,ds,\tau_t h\right). \tag*{(22)} J S t J − 1 ( a , h ) = ( a + ∫ 0 t h ( s ) d s , τ t h ) . (22) The first component converges to a a a h h h h h h H γ H_\gamma H γ ( S t ) t ≥ 0 (S_t)_{t\ge0} ( S t ) t ≥ 0
It remains to identify the generator. Suppose first that θ ′ ∈ E γ \theta'\in E_\gamma θ ′ ∈ E γ h = θ ′ h=\theta' h = θ ′ h h h h ′ ∈ H γ h'\in H_\gamma h ′ ∈ H γ h ( 0 ) h(0) h ( 0 )
J S t θ − θ t = ( 1 t ∫ 0 t h ( s ) d s , τ t h − h t ) . \frac{JS_t\theta-\theta}{t}=\left(\frac{1}{t}\int_{0}^{t}h(s)\,ds,\frac{\tau_t h-h}{t}\right). t J S t θ − θ = ( t 1 ∫ 0 t h ( s ) d s , t τ t h − h ) . The first component converges to h ( 0 ) h(0) h ( 0 )
τ t h − h t − h ′ = ∫ 0 1 ( τ t v h ′ − h ′ ) d v . \frac{\tau_t h-h}{t}-h'=\int_{0}^{1}(\tau_{tv}h'-h')\,dv. t τ t h − h − h ′ = ∫ 0 1 ( τ t v h ′ − h ′ ) d v . In particular,
∥ τ t h − h t − h ′ ∥ H γ ≤ sup 0 ≤ s ≤ t ∥ τ s h ′ − h ′ ∥ H γ ⟶ 0 \left\|\frac{\tau_t h-h}{t}-h'\right\|_{H_\gamma}\le\sup_{0\le s\le t}\|\tau_s h'-h'\|_{H_\gamma}\longrightarrow0 t τ t h − h − h ′ H γ ≤ 0 ≤ s ≤ t sup ∥ τ s h ′ − h ′ ∥ H γ ⟶ 0 by strong continuity at zero. Thus ( S t θ − θ ) / t → θ ′ (S_t\theta-\theta)/t\to\theta' ( S t θ − θ ) / t → θ ′ E γ E_\gamma E γ
Conversely, suppose that ( S t θ − θ ) / t (S_t\theta-\theta)/t ( S t θ − θ ) / t E γ E_\gamma E γ t ↓ 0 t\downarrow0 t ↓ 0 h = θ ′ h=\theta' h = θ ′ ( τ t h − h ) / t → g (\tau_t h-h)/t\to g ( τ t h − h ) / t → g H γ H_\gamma H γ g g g ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) L l o c 2 L^2_{\mathrm{loc}} L loc 2 φ ∈ C c ∞ ( ( 0 , ∞ ) ) \varphi\in C_c^\infty((0,\infty)) φ ∈ C c ∞ (( 0 , ∞ ))
∫ 0 ∞ g ( r ) φ ( r ) d r = lim t ↓ 0 ∫ 0 ∞ h ( r + t ) − h ( r ) t φ ( r ) d r = − ∫ 0 ∞ h ( r ) φ ′ ( r ) d r . \int_{0}^{\infty}g(r)\varphi(r)\,dr=\lim_{t\downarrow0}\int_{0}^{\infty}\frac{h(r+t)-h(r)}{t}\varphi(r)\,dr=-\int_{0}^{\infty}h(r)\varphi'(r)\,dr. ∫ 0 ∞ g ( r ) φ ( r ) d r = t ↓ 0 lim ∫ 0 ∞ t h ( r + t ) − h ( r ) φ ( r ) d r = − ∫ 0 ∞ h ( r ) φ ′ ( r ) d r . Hence g g g h h h ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) h h h g g g H γ H_\gamma H γ w γ − 1 w_\gamma^{-1} w γ − 1
∫ 0 T ( ∣ h ( r ) ∣ + ∣ g ( r ) ∣ ) d r ≤ J γ ( T ) 1 / 2 ( ∥ h ∥ H γ + ∥ g ∥ H γ ) < ∞ . \int_{0}^{T}(|h(r)|+|g(r)|)\,dr\le J_\gamma(T)^{1/2}(\|h\|_{H_\gamma}+\|g\|_{H_\gamma})<\infty. ∫ 0 T ( ∣ h ( r ) ∣ + ∣ g ( r ) ∣ ) d r ≤ J γ ( T ) 1/2 ( ∥ h ∥ H γ + ∥ g ∥ H γ ) < ∞. To see the behavior at the origin explicitly, fix r 0 > 0 r_0>0 r 0 > 0 h h h ( 0 , r 0 ] (0,r_0] ( 0 , r 0 ] h ( r ) = h ( r 0 ) − ∫ r r 0 g ( q ) d q h(r)=h(r_0)-\int_{r}^{r_0}g(q)\,dq h ( r ) = h ( r 0 ) − ∫ r r 0 g ( q ) d q g g g r ↓ 0 r\downarrow0 r ↓ 0 [ 0 , r 0 ] [0,r_0] [ 0 , r 0 ] h ∈ A C l o c ( R + ) h\in AC_{\mathrm{loc}}(\mathbb{R}_+) h ∈ A C loc ( R + ) ∣ h ( 0 ) ∣ 2 + ∥ g ∥ H γ 2 < ∞ |h(0)|^2+\|g\|_{H_\gamma}^2<\infty ∣ h ( 0 ) ∣ 2 + ∥ g ∥ H γ 2 < ∞ h ∈ E γ h\in E_\gamma h ∈ E γ t − 1 ∫ 0 t h ( s ) d s t^{-1}\int_{0}^{t}h(s)\,ds t − 1 ∫ 0 t h ( s ) d s h ( 0 ) h(0) h ( 0 ) h = θ ′ h=\theta' h = θ ′
For h ∈ U b h \in U_b h ∈ U b
( G b h ) ( r ) : = ⟨ ϕ r , h ⟩ U b . (23) (G_bh)(r) := \langle\phi_r,h\rangle_{U_b}. \tag*{(23)} ( G b h ) ( r ) := ⟨ ϕ r , h ⟩ U b . (23) For r > 0 r > 0 r > 0
ϕ r ′ ( λ ) = b e − λ r , \phi'_r(\lambda) = \sqrt{b}e^{-\lambda r}, ϕ r ′ ( λ ) = b e − λ r , and hence
∥ ϕ r ′ ∥ U b 2 = b Γ ( 2 − b ) ∫ 0 ∞ e − 2 λ r λ 1 − b d λ = b ( 2 r ) b − 2 . (24) \|\phi'_r\|_{U_b}^2 = \frac{b}{\Gamma(2-b)}\int_0^\infty e^{-2\lambda r}\lambda^{1-b}\,d\lambda= b(2r)^{b-2}. \tag*{(24)} ∥ ϕ r ′ ∥ U b 2 = Γ ( 2 − b ) b ∫ 0 ∞ e − 2 λ r λ 1 − b d λ = b ( 2 r ) b − 2 . (24) For every T > 0 T > 0 T > 0 h ∈ U b h \in U_b h ∈ U b
∫ 0 T ∣ ⟨ ϕ r ′ , h ⟩ U b ∣ d r ≤ ∥ h ∥ U b ∫ 0 T ∥ ϕ r ′ ∥ U b d r = ∥ h ∥ U b b 2 ( b − 2 ) / 2 ∫ 0 T r ( b − 2 ) / 2 d r < ∞ , \begin{aligned}
\int_0^T |\langle\phi'_r,h\rangle_{U_b}|\,dr \le\|h\|_{U_b}\int_0^T \|\phi'_r\|_{U_b}\,dr \\
&= \|h\|_{U_b}\sqrt{b}\,2^{(b-2)/2}\int_0^T r^{(b-2)/2}\,dr < \infty,
\end{aligned} ∫ 0 T ∣ ⟨ ϕ r ′ , h ⟩ U b ∣ d r ≤ ∥ h ∥ U b ∫ 0 T ∥ ϕ r ′ ∥ U b d r = ∥ h ∥ U b b 2 ( b − 2 ) /2 ∫ 0 T r ( b − 2 ) /2 d r < ∞ , because b > 0 b > 0 b > 0 ϕ r = ∫ 0 r ϕ q ′ d q \phi_r = \int_0^r \phi'_q\,dq ϕ r = ∫ 0 r ϕ q ′ d q U b U_b U b G b h ∈ A C l o c ( R + ) G_bh \in AC_{\mathrm{loc}}(\mathbb{R}_+) G b h ∈ A C loc ( R + )
( G b h ) ′ ( r ) = ⟨ ϕ r ′ , h ⟩ U b (G_bh)'(r) = \langle\phi'_r,h\rangle_{U_b} ( G b h ) ′ ( r ) = ⟨ ϕ r ′ , h ⟩ U b for almost every r > 0 r > 0 r > 0
Lemma 2.4. Let 0 < b < 2 0 < b < 2 0 < b < 2 0 ≤ γ < 1 0 \le\gamma< 1 0 ≤ γ < 1 G b ∈ L 2 ( U b , E γ ) G_b \in\mathcal{L}_2(U_b,\mathcal{E}_\gamma) G b ∈ L 2 ( U b , E γ )
b + γ > 1. (25) b+\gamma> 1. \tag*{(25)} b + γ > 1. (25) In that case,
∥ G b ∥ H S ( U b , E γ ) 2 = b 2 b − 2 Γ ( b + γ − 1 ) Γ ( 3 − b − γ ) . (26) \|G_b\|_{\mathrm{HS}(U_b,\mathcal{E}_\gamma)}^2 = b2^{b-2}\Gamma(b+\gamma-1)\Gamma(3-b-\gamma). \tag*{(26)} ∥ G b ∥ HS ( U b , E γ ) 2 = b 2 b − 2 Γ ( b + γ − 1 ) Γ ( 3 − b − γ ) . (26) Proof. If b + γ > 1 b+\gamma> 1 b + γ > 1
∥ G b h ∥ γ 2 ≤ b 2 b − 2 ∥ h ∥ U b 2 ∫ 0 ∞ r b + γ − 2 ( 1 + r ) 2 d r < ∞ , \|G_bh\|_\gamma^2 \le b2^{b-2}\|h\|_{U_b}^2\int_0^\infty\frac{r^{b+\gamma-2}}{(1+r)^2}\,dr < \infty, ∥ G b h ∥ γ 2 ≤ b 2 b − 2 ∥ h ∥ U b 2 ∫ 0 ∞ ( 1 + r ) 2 r b + γ − 2 d r < ∞ , so (23) maps U b U_b U b E γ \mathcal{E}_\gamma E γ ( e n ) (e_n) ( e n ) U b U_b U b ϕ 0 = 0 \phi_0=0 ϕ 0 = 0
∑ n = 1 ∞ ∥ G b e n ∥ γ 2 = ∫ 0 ∞ r γ ( 1 + r ) 2 ∑ n = 1 ∞ ∣ ⟨ ϕ r ′ , e n ⟩ U b ∣ 2 d r = b 2 b − 2 ∫ 0 ∞ r b + γ − 2 ( 1 + r ) 2 d r . \begin{aligned}
\sum_{n=1}^\infty\|G_be_n\|_\gamma^2 = \int_0^\infty\frac{r^\gamma}{(1+r)^2}\sum_{n=1}^\infty|\langle\phi'_r,e_n\rangle_{U_b}|^2\,dr \\
&= b2^{b-2}\int_0^\infty\frac{r^{b+\gamma-2}}{(1+r)^2}\,dr.
\end{aligned} n = 1 ∑ ∞ ∥ G b e n ∥ γ 2 = ∫ 0 ∞ ( 1 + r ) 2 r γ n = 1 ∑ ∞ ∣ ⟨ ϕ r ′ , e n ⟩ U b ∣ 2 d r = b 2 b − 2 ∫ 0 ∞ ( 1 + r ) 2 r b + γ − 2 d r . Near zero the integrand is equivalent to r b + γ − 2 r^{b+\gamma-2} r b + γ − 2 b + γ > 1 b+\gamma> 1 b + γ > 1 r b + γ − 4 r^{b+\gamma-4} r b + γ − 4 b < 2 b < 2 b < 2 γ < 1 \gamma< 1 γ < 1 b + γ > 1 b+\gamma> 1 b + γ > 1 a = b + γ − 1 ∈ ( 0 , 2 ) a=b+\gamma-1 \in(0,2) a = b + γ − 1 ∈ ( 0 , 2 )
∫ 0 ∞ r a − 1 ( 1 + r ) 2 d r = ∫ 0 1 v a − 1 ( 1 − v ) 1 − a d v = Γ ( a ) Γ ( 2 − a ) , \int_0^\infty\frac{r^{a-1}}{(1+r)^2}\,dr = \int_0^1 v^{a-1}(1-v)^{1-a}\,dv = \Gamma(a)\Gamma(2-a), ∫ 0 ∞ ( 1 + r ) 2 r a − 1 d r = ∫ 0 1 v a − 1 ( 1 − v ) 1 − a d v = Γ ( a ) Γ ( 2 − a ) , where v = r / ( 1 + r ) v=r/(1+r) v = r / ( 1 + r ) E γ \mathcal{E}_\gamma E γ b + γ ≤ 1 b+\gamma\le1 b + γ ≤ 1
Remark 2.5 For b = 1 b=1 b = 1 μ 1 ( d λ ) = d λ \mu_1(\mathrm{d}\lambda)=\mathrm{d}\lambda μ 1 ( d λ ) = d λ φ r ′ ( λ ) = e − λ r \varphi_r^\prime(\lambda)=e^{-\lambda r} φ r ′ ( λ ) = e − λ r
∥ φ r ′ ∥ 1 2 = ∫ 0 ∞ e − 2 λ r d λ = 1 2 r . \|\varphi_r^\prime\|_{1}^{2}=\int_{0}^{\infty}e^{-2\lambda r}\,\mathrm{d}\lambda=\frac{1}{2r}. ∥ φ r ′ ∥ 1 2 = ∫ 0 ∞ e − 2 λ r d λ = 2 r 1 . The condition in Lemma 2.4 reduces to γ > 0 \gamma>0 γ > 0 0 < γ < 1 0<\gamma<1 0 < γ < 1
∥ G 1 ∥ H S ( U 1 , E γ ) 2 = 1 2 Γ ( γ ) Γ ( 2 − γ ) = π ( 1 − γ ) 2 sin ( π γ ) < ∞ . \|G_1\|_{\mathrm{HS}(U_1,E_\gamma)}^{2}=\frac{1}{2}\Gamma(\gamma)\Gamma(2-\gamma)=\frac{\pi(1-\gamma)}{2\sin(\pi\gamma)}<\infty. ∥ G 1 ∥ HS ( U 1 , E γ ) 2 = 2 1 Γ ( γ ) Γ ( 2 − γ ) = 2 sin ( π γ ) π ( 1 − γ ) < ∞. At γ = 0 \gamma=0 γ = 0 ∫ 0 ∞ r − 1 ( 1 + r ) − 2 d r \int_{0}^{\infty}r^{-1}(1+r)^{-2}\,\mathrm{d}r ∫ 0 ∞ r − 1 ( 1 + r ) − 2 d r r − 1 ( 1 + r ) − 2 ∼ r − 1 r^{-1}(1+r)^{-2}\sim r^{-1} r − 1 ( 1 + r ) − 2 ∼ r − 1 r ↓ 0 r\downarrow0 r ↓ 0 G 1 ∉ L 2 ( U 1 , E 0 ) G_1\notin\mathcal{L}_2(U_1,E_0) G 1 ∈ / L 2 ( U 1 , E 0 ) γ = 0 \gamma=0 γ = 0 1 < b < 2 1<b<2 1 < b < 2 γ \gamma γ
Remark 2.6 More generally, the derivative weight w γ , m ( r ) = r γ / ( 1 + r ) m w_{\gamma,m}(r)=r^\gamma/(1+r)^m w γ , m ( r ) = r γ / ( 1 + r ) m ∫ 0 ∞ r b + γ − 2 ( 1 + r ) − m d r \int_{0}^{\infty}r^{b+\gamma-2}(1+r)^{-m}\,\mathrm{d}r ∫ 0 ∞ r b + γ − 2 ( 1 + r ) − m d r 1 < b + γ < m + 1 1<b+\gamma<m+1 1 < b + γ < m + 1 m = 2 m=2 m = 2 0 < b < 2 0<b<2 0 < b < 2 0 ≤ γ < 1 0\leq\gamma<1 0 ≤ γ < 1
The transport lift
The term transport refers to right translation in the horizon variable r r r A θ = θ ′ A\theta=\theta^\prime A θ = θ ′
Fix an orthonormal basis ( e k ) k ≥ 1 (e_k)_{k\geq1} ( e k ) k ≥ 1 U b U_b U b ( F t ) (\mathcal{F}_t) ( F t ) ( β k ) k ≥ 1 (\beta_k)_{k\geq1} ( β k ) k ≥ 1 W W W U b U_b U b
W t ( h ) : = ∑ k = 1 ∞ ⟨ h , e k ⟩ U b β k ( t ) , h ∈ U b , W_t(h):=\sum_{k=1}^{\infty}\langle h,e_k\rangle_{U_b}\,\beta_k(t),\qquad h\in U_b, W t ( h ) := k = 1 ∑ ∞ ⟨ h , e k ⟩ U b β k ( t ) , h ∈ U b , where the series converges in L 2 ( Ω ) L^2(\Omega) L 2 ( Ω ) E [ W t ( h ) W s ( g ) ] = ( t ∧ s ) ⟨ h , g ⟩ U b \mathbb{E}[W_t(h)W_s(g)]=(t\wedge s)\langle h,g\rangle_{U_b} E [ W t ( h ) W s ( g )] = ( t ∧ s ) ⟨ h , g ⟩ U b
The Hilbert–Schmidt condition makes the variance of the Brownian series summable in the state norm. For a finite partial sum, the Itô isometry gives
E ∥ ∑ k = 1 n G b e k β k ( t ) ∥ γ 2 = t ∑ k = 1 n ∥ G b e k ∥ γ 2 . \mathbb{E}\left\|\sum_{k=1}^{n}G_be_k\beta_k(t)\right\|_{\gamma}^{2}=t\sum_{k=1}^{n}\|G_be_k\|_{\gamma}^{2}. E k = 1 ∑ n G b e k β k ( t ) γ 2 = t k = 1 ∑ n ∥ G b e k ∥ γ 2 . The series therefore converges in L 2 ( Ω ; E γ ) L^2(\Omega;E_\gamma) L 2 ( Ω ; E γ )
With this realization, ζ t , f , b \zeta_{t,f,b} ζ t , f , b
ζ t , f , b = ∑ k = 1 ∞ ∫ 0 t f ( u ) ⟨ φ t − u , e k ⟩ U b d β k ( u ) . (27) \zeta_{t,f,b}=\sum_{k=1}^{\infty}\int_{0}^{t}\sqrt{f(u)}\langle\varphi_{t-u},e_k\rangle_{U_b}\,\mathrm{d}\beta_k(u). \tag*{(27)} ζ t , f , b = k = 1 ∑ ∞ ∫ 0 t f ( u ) ⟨ φ t − u , e k ⟩ U b d β k ( u ) . (27) where the series converges in L 2 ( Ω ) L^2(\Omega) L 2 ( Ω ) E E E K ∈ L 2 ( 0 , t ; L 2 ( U b , E ) ) K\in L^2(0,t;\mathcal{L}_2(U_b,E)) K ∈ L 2 ( 0 , t ; L 2 ( U b , E ))
∫ 0 t K u d W u : = ∑ k = 1 ∞ ∫ 0 t K u e k d β k ( u ) , \int_{0}^{t}K_u\,\mathrm{d}W_u:=\sum_{k=1}^{\infty}\int_{0}^{t}K_ue_k\,\mathrm{d}\beta_k(u), ∫ 0 t K u d W u := k = 1 ∑ ∞ ∫ 0 t K u e k d β k ( u ) , with convergence in L 2 ( Ω ; E ) L^2(\Omega;E) L 2 ( Ω ; E )
Indeed, the scalar Itô isometry and independence of the Brownian coordinates give, for n > m n>m n > m
E ∥ ∑ k = m + 1 n ∫ 0 t K u e k d β k ( u ) ∥ E 2 = ∫ 0 t ∑ k = m + 1 n ∥ K u e k ∥ E 2 d u ⟶ 0. \mathbb{E}\left\|\sum_{k=m+1}^{n}\int_{0}^{t}K_{u}e_{k}\,d\beta_{k}(u)\right\|_{E}^{2}=\int_{0}^{t}\sum_{k=m+1}^{n}\left\|K_{u}e_{k}\right\|_{E}^{2}\,du\longrightarrow0. E k = m + 1 ∑ n ∫ 0 t K u e k d β k ( u ) E 2 = ∫ 0 t k = m + 1 ∑ n ∥ K u e k ∥ E 2 d u ⟶ 0. This also gives the Hilbert-space Itô isometry E ∥ ∫ 0 t K u d W u ∥ E 2 = ∫ 0 t ∥ K u ∥ H S 2 d u \mathbb{E}\left\|\int_{0}^{t}K_{u}\,dW_{u}\right\|_{E}^{2}=\int_{0}^{t}\left\|K_{u}\right\|_{\mathrm{HS}}^{2}\,du E ∫ 0 t K u d W u E 2 = ∫ 0 t ∥ K u ∥ HS 2 d u K u = S t − u f ( u ) G b K_{u}=S_{t-u}\sqrt{f(u)}G_{b} K u = S t − u f ( u ) G b
Theorem 2.7 Let 0 < b < 2 0<b<2 0 < b < 2 0 < γ < 1 0<\gamma<1 0 < γ < 1 b + γ > 1 b+\gamma>1 b + γ > 1 f ∈ L l o c 1 ( R + ) f\in L^{1}_{\mathrm{loc}}(\mathbb{R}_{+}) f ∈ L loc 1 ( R + )
d Θ t = A Θ t d t + f ( t ) G b d W t , Θ 0 = 0 , (28) d\Theta_{t}=A\Theta_{t}\,dt+\sqrt{f(t)}G_{b}\,dW_{t},\qquad\Theta_{0}=0, \tag*{(28)} d Θ t = A Θ t d t + f ( t ) G b d W t , Θ 0 = 0 , (28) where the cylindrical noise is understood through the Brownian coordinates ( β k ) (\beta_{k}) ( β k )
Θ t = ∑ k = 1 ∞ ∫ 0 t S t − u f ( u ) G b e k d β k ( u ) in L 2 ( Ω ; E γ ) . (29) \Theta_{t}=\sum_{k=1}^{\infty}\int_{0}^{t}S_{t-u}\sqrt{f(u)}G_{b}e_{k}\,d\beta_{k}(u)\quad\text{in }L^{2}(\Omega;E_{\gamma}). \tag*{(29)} Θ t = k = 1 ∑ ∞ ∫ 0 t S t − u f ( u ) G b e k d β k ( u ) in L 2 ( Ω ; E γ ) . (29) For every r ≥ 0 r\geq0 r ≥ 0
Θ t ( r ) = ∑ k = 1 ∞ ∫ 0 t f ( u ) ⟨ ϕ t + r − u , e k ⟩ U b d β k ( u ) , (30) \Theta_{t}(r)=\sum_{k=1}^{\infty}\int_{0}^{t}\sqrt{f(u)}\langle\phi_{t+r-u},e_{k}\rangle_{U_{b}}\,d\beta_{k}(u), \tag*{(30)} Θ t ( r ) = k = 1 ∑ ∞ ∫ 0 t f ( u ) ⟨ ϕ t + r − u , e k ⟩ U b d β k ( u ) , (30) and the boundary process satisfies
δ 0 Θ t = Θ t ( 0 ) = ζ t , f , b . (31) \delta_{0}\Theta_{t}=\Theta_{t}(0)=\zeta_{t,f,b}. \tag*{(31)} δ 0 Θ t = Θ t ( 0 ) = ζ t , f , b . (31) Moreover,
Θ t + h ( r ) = Θ t ( r + h ) + ∑ k = 1 ∞ ∫ t t + h f ( u ) ⟨ ϕ t + h + r − u , e k ⟩ U b d β k ( u ) . (32) \Theta_{t+h}(r)=\Theta_{t}(r+h)+\sum_{k=1}^{\infty}\int_{t}^{t+h}\sqrt{f(u)}\langle\phi_{t+h+r-u},e_{k}\rangle_{U_{b}}\,d\beta_{k}(u). \tag*{(32)} Θ t + h ( r ) = Θ t ( r + h ) + k = 1 ∑ ∞ ∫ t t + h f ( u ) ⟨ ϕ t + h + r − u , e k ⟩ U b d β k ( u ) . (32) The flow identity defines a time-inhomogeneous Gaussian Markov process with a Feller transition evolution on E γ E_{\gamma} E γ f ∈ L l o c ∞ ( R + ) f\in L^{\infty}_{\mathrm{loc}}(\mathbb{R}_{+}) f ∈ L loc ∞ ( R + ) E γ E_{\gamma} E γ
Proof Fix T > 0 T>0 T > 0 E : = E γ E:=E_{\gamma} E := E γ t ≤ T t\leq T t ≤ T
∫ 0 t f ( u ) ∥ S t − u G b ∥ H S ( U b , E ) 2 d u ≤ C γ , T 2 ∥ G b ∥ H S ( U b , E ) 2 ∥ f ∥ L 1 ( 0 , T ) < ∞ . \int_{0}^{t}f(u)\left\|S_{t-u}G_{b}\right\|_{\mathrm{HS}(U_{b},E)}^{2}\,du\leq C_{\gamma,T}^{2}\left\|G_{b}\right\|_{\mathrm{HS}(U_{b},E)}^{2}\left\|f\right\|_{L^{1}(0,T)}<\infty. ∫ 0 t f ( u ) ∥ S t − u G b ∥ HS ( U b , E ) 2 d u ≤ C γ , T 2 ∥ G b ∥ HS ( U b , E ) 2 ∥ f ∥ L 1 ( 0 , T ) < ∞. Thus u ↦ S t − u f ( u ) G b u\mapsto S_{t-u}\sqrt{f(u)}G_{b} u ↦ S t − u f ( u ) G b L 2 ( 0 , t ; L 2 ( U b , E ) ) L^{2}(0,t;L_{2}(U_{b},E)) L 2 ( 0 , t ; L 2 ( U b , E )) E E E
The map t ↦ Θ t t\mapsto\Theta_{t} t ↦ Θ t [ 0 , T ] [0,T] [ 0 , T ] L 2 ( Ω ; E ) L^{2}(\Omega;E) L 2 ( Ω ; E ) [ 0 , T ] [0,T] [ 0 , T ] L 2 ( 0 , T ; L 2 ( U b , E ) ) L^2(0,T;\mathcal{L}_2(U_b,E)) L 2 ( 0 , T ; L 2 ( U b , E )) S S S S t n − u G b → S t − u G b S_{t_n-u}G_b \to S_{t-u}G_b S t n − u G b → S t − u G b
∥ ( S t n − u − S t − u ) G b ∥ H S 2 = ∑ j ∥ ( S t n − u − S t − u ) G b e j ∥ E 2 → 0. \lVert(S_{t_n-u}-S_{t-u})G_b\rVert_{\mathrm{HS}}^2=\sum_j\lVert(S_{t_n-u}-S_{t-u})G_be_j\rVert_E^2\to0. ∥( S t n − u − S t − u ) G b ∥ HS 2 = j ∑ ∥( S t n − u − S t − u ) G b e j ∥ E 2 → 0. by dominated convergence over an orthonormal basis ( e j ) (e_j) ( e j ) f ∈ L 1 ( 0 , T ) f\in L^1(0,T) f ∈ L 1 ( 0 , T )
We construct a predictable modification directly. The L 2 ( Ω ; E ) L^2(\Omega;E) L 2 ( Ω ; E ) [ 0 , T ] [0,T] [ 0 , T ] Y n ( t ) Y_n(t) Y n ( t ) Θ t i \Theta_{t_i} Θ t i ( t i , t i + 1 ] (t_i,t_{i+1}] ( t i , t i + 1 ] Y n ( 0 ) = 0 Y_n(0)=0 Y n ( 0 ) = 0
sup t ≤ T E ∥ Y n ( t ) − Θ t ∥ E 2 ≤ 2 − 3 n . \sup_{t\leq T}\mathbb{E}\lVert Y_n(t)-\Theta_t\rVert_E^2\leq2^{-3n}. t ≤ T sup E ∥ Y n ( t ) − Θ t ∥ E 2 ≤ 2 − 3 n . Each Y n Y_n Y n Θ t i \Theta_{t_i} Θ t i F t i \mathcal{F}_{t_i} F t i t t t ε > 0 \varepsilon>0 ε > 0
∑ n = 1 ∞ P ( ∥ Y n ( t ) − Θ t ∥ E > ε ) ≤ ε − 2 ∑ n = 1 ∞ 2 − 3 n < ∞ . \sum_{n=1}^{\infty}\mathbb{P}(\lVert Y_n(t)-\Theta_t\rVert_E>\varepsilon)\leq\varepsilon^{-2}\sum_{n=1}^{\infty}2^{-3n}<\infty. n = 1 ∑ ∞ P (∥ Y n ( t ) − Θ t ∥ E > ε ) ≤ ε − 2 n = 1 ∑ ∞ 2 − 3 n < ∞. The Borel–Cantelli lemma, applied with ε = 1 / m \varepsilon=1/m ε = 1/ m m ∈ N m\in\mathbb{N} m ∈ N Y n ( t ) → Θ t Y_n(t)\to\Theta_t Y n ( t ) → Θ t E E E Θ \Theta Θ R + \mathbb{R}_+ R +
For every r ≥ 0 r\geq0 r ≥ 0 δ r θ : = θ ( r ) \delta_r\theta:=\theta(r) δ r θ := θ ( r ) E E E δ r = δ 0 S r \delta_r=\delta_0S_r δ r = δ 0 S r
δ r S t − u G b h = ( G b h ) ( t − u + r ) = ⟨ φ t − u + r , h ⟩ U b . \delta_rS_{t-u}G_bh=(G_bh)(t-u+r)=\langle\varphi_{t-u+r},h\rangle_{U_b}. δ r S t − u G b h = ( G b h ) ( t − u + r ) = ⟨ φ t − u + r , h ⟩ U b . Applying the bounded functional δ r \delta_r δ r
E [ Θ s ( 0 ) Θ t ( 0 ) ] = ∫ 0 s ∧ t f ( u ) ⟨ φ s − u , φ t − u ⟩ U b d u = R f , b ( s , t ) . \mathbb{E}[\Theta_s(0)\Theta_t(0)]=\int_0^{s\wedge t}f(u)\langle\varphi_{s-u},\varphi_{t-u}\rangle_{U_b}\,\mathrm{d}u=R_{f,b}(s,t). E [ Θ s ( 0 ) Θ t ( 0 )] = ∫ 0 s ∧ t f ( u ) ⟨ φ s − u , φ t − u ⟩ U b d u = R f , b ( s , t ) . Thus the boundary in (31) is a realization of the Gaussian process defined by (1).
Assume now that f ∈ L ∞ ( 0 , T ) f\in L^\infty(0,T) f ∈ L ∞ ( 0 , T ) p > 2 p>2 p > 2 a ∈ ( 1 / p , 1 / 2 ) a\in(1/p,1/2) a ∈ ( 1/ p , 1/2 )
sup t ≤ T ∫ 0 t ( t − u ) − 2 a f ( u ) ∥ S t − u G b ∥ H S 2 d u ≤ C γ , T 2 ∥ f ∥ ∞ , T ∥ G b ∥ H S 2 T 1 − 2 a 1 − 2 a < ∞ . (33) \sup_{t\leq T}\int_0^t(t-u)^{-2a}f(u)\lVert S_{t-u}G_b\rVert_{\mathrm{HS}}^2\,\mathrm{d}u\leq\frac{C_{\gamma,T}^2\lVert f\rVert_{\infty,T}\lVert G_b\rVert_{\mathrm{HS}}^2T^{1-2a}}{1-2a}<\infty. \tag*{(33)} t ≤ T sup ∫ 0 t ( t − u ) − 2 a f ( u ) ∥ S t − u G b ∥ HS 2 d u ≤ 1 − 2 a C γ , T 2 ∥ f ∥ ∞ , T ∥ G b ∥ HS 2 T 1 − 2 a < ∞. (33) Define
Z a ( t ) : = ∑ k = 1 ∞ ∫ 0 t ( t − u ) − a S t − u f ( u ) G b e k d β k ( u ) . Z_a(t):=\sum_{k=1}^{\infty}\int_0^t(t-u)^{-a}S_{t-u}\sqrt{f(u)}G_be_k\,\mathrm{d}\beta_k(u). Z a ( t ) := k = 1 ∑ ∞ ∫ 0 t ( t − u ) − a S t − u f ( u ) G b e k d β k ( u ) . The deterministic kernels belong to L 2 ( { ( t , u ) : 0 < u < t < T } ; L 2 ( U b , E ) ) L^2(\{(t,u):0<u<t<T\};\mathcal{L}_2(U_b,E)) L 2 ({( t , u ) : 0 < u < t < T } ; L 2 ( U b , E )) Z a Z_a Z a E E E Z Z Z q = E ∥ Z ∥ E 2 q=\mathbb{E}\lVert Z\rVert_E^2 q = E ∥ Z ∥ E 2 ∥ Z ∥ E 2 = ∑ j q j ξ j 2 \lVert Z\rVert_E^2=\sum_jq_j\xi_j^2 ∥ Z ∥ E 2 = ∑ j q j ξ j 2 q j ≥ 0 q_j\geq0 q j ≥ 0 ∑ j q j = q \sum_jq_j=q ∑ j q j = q ξ j \xi_j ξ j q > 0 q>0 q > 0
E ∥ Z ∥ E p ≤ q p / 2 ∑ j q j q E ∣ ξ j ∣ p = E ∣ ξ 1 ∣ p q p / 2 ; \mathbb{E}\lVert Z\rVert_E^p\leq q^{p/2}\sum_j\frac{q_j}{q}\mathbb{E}|\xi_j|^p=\mathbb{E}|\xi_1|^p q^{p/2}; E ∥ Z ∥ E p ≤ q p /2 j ∑ q q j E ∣ ξ j ∣ p = E ∣ ξ 1 ∣ p q p /2 ; the case q = 0 q=0 q = 0 Z a ( t ) Z_a(t) Z a ( t )
E ∥ Z a ( t ) ∥ E p ≤ C p ( E ∥ Z a ( t ) ∥ E 2 ) p / 2 . \mathbb{E}\lVert Z_a(t)\rVert_E^p\leq C_p\bigl(\mathbb{E}\lVert Z_a(t)\rVert_E^2\bigr)^{p/2}. E ∥ Z a ( t ) ∥ E p ≤ C p ( E ∥ Z a ( t ) ∥ E 2 ) p /2 . Together with (33) and Tonelli’s theorem, this shows that Z a ∈ L p ( Ω ; L p ( 0 , T ; E ) ) Z_a \in L^p(\Omega;L^p(0,T;E)) Z a ∈ L p ( Ω ; L p ( 0 , T ; E ))
∫ 0 t ( t − v ) a − 1 ( ∫ 0 v ( v − u ) − 2 a f ( u ) ∥ S t − u G b ∥ H S 2 d u ) 1 / 2 d v ≤ C ∫ 0 t ( t − v ) a − 1 v 1 / 2 − a d v < ∞ . \int_0^t (t-v)^{a-1}\left(\int_0^v (v-u)^{-2a}f(u)\lVert S_{t-u}G_b\rVert_{\mathrm{HS}}^2\,\mathrm{d}u\right)^{1/2}\,\mathrm{d}v
\le C\int_0^t (t-v)^{a-1}v^{1/2-a}\,\mathrm{d}v<\infty. ∫ 0 t ( t − v ) a − 1 ( ∫ 0 v ( v − u ) − 2 a f ( u ) ∥ S t − u G b ∥ HS 2 d u ) 1/2 d v ≤ C ∫ 0 t ( t − v ) a − 1 v 1/2 − a d v < ∞. For fixed t t t v v v L 2 ( 0 , t ; L 2 ( U b , E ) ) L^2(0,t;L_2(U_b,E)) L 2 ( 0 , t ; L 2 ( U b , E )) L 2 ( Ω ; E ) L^2(\Omega;E) L 2 ( Ω ; E )
∫ 0 t ( t − v ) a − 1 S t − v Z a ( v ) d v = ∫ 0 t ( ∫ u t ( t − v ) a − 1 ( v − u ) − a d v ) S t − u f ( u ) G b d W u = π sin ( π a ) Θ t . \begin{aligned}
\int_0^t (t-v)^{a-1}S_{t-v}Z_a(v)\,\mathrm{d}v \\
&= \int_0^t\left(\int_u^t (t-v)^{a-1}(v-u)^{-a}\,\mathrm{d}v\right)S_{t-u}\sqrt{f(u)}G_b\,\mathrm{d}W_u \\
&= \frac{\pi}{\sin(\pi a)}\Theta_t.
\end{aligned} ∫ 0 t ( t − v ) a − 1 S t − v Z a ( v ) d v = ∫ 0 t ( ∫ u t ( t − v ) a − 1 ( v − u ) − a d v ) S t − u f ( u ) G b d W u = sin ( π a ) π Θ t . The inner integral equals ∫ 0 1 ( 1 − z ) a − 1 z − a d z = Γ ( a ) Γ ( 1 − a ) = π / sin ( π a ) \int_0^1(1-z)^{a-1}z^{-a}\,\mathrm{d}z=\Gamma(a)\Gamma(1-a)=\pi/\sin(\pi a) ∫ 0 1 ( 1 − z ) a − 1 z − a d z = Γ ( a ) Γ ( 1 − a ) = π / sin ( π a ) z = ( v − u ) / ( t − u ) z=(v-u)/(t-u) z = ( v − u ) / ( t − u )
For completeness, define
( R a z ) ( t ) : = ∫ 0 t ( t − v ) a − 1 S t − v z ( v ) d v . (R_a z)(t):=\int_0^t (t-v)^{a-1}S_{t-v}z(v)\,\mathrm{d}v. ( R a z ) ( t ) := ∫ 0 t ( t − v ) a − 1 S t − v z ( v ) d v . Let q = p / ( p − 1 ) q=p/(p-1) q = p / ( p − 1 ) a > 1 / p a>1/p a > 1/ p ( a − 1 ) q > − 1 (a-1)q>-1 ( a − 1 ) q > − 1
sup t ≤ T ∥ ( R a z ) ( t ) ∥ E ≤ C γ , T ( T ( a − 1 ) q + 1 ( a − 1 ) q + 1 ) 1 / q ∥ z ∥ L p ( 0 , T ; E ) . \sup_{t\le T}\lVert(R_a z)(t)\rVert_E\le C_{\gamma,T}\left(\frac{T^{(a-1)q+1}}{(a-1)q+1}\right)^{1/q}\lVert z\rVert_{L^p(0,T;E)}. t ≤ T sup ∥( R a z ) ( t ) ∥ E ≤ C γ , T ( ( a − 1 ) q + 1 T ( a − 1 ) q + 1 ) 1/ q ∥ z ∥ L p ( 0 , T ; E ) . If z z z R a z R_a z R a z ∫ 0 t r a − 1 S r z ( t − r ) d r \int_0^t r^{a-1}S_rz(t-r)\,\mathrm{d}r ∫ 0 t r a − 1 S r z ( t − r ) d r [ 0 , T ] [0,T] [ 0 , T ] r a − 1 r^{a-1} r a − 1 E E E L p ( 0 , T ; E ) L^p(0,T;E) L p ( 0 , T ; E ) z ∈ L p ( 0 , T ; E ) z\in L^p(0,T;E) z ∈ L p ( 0 , T ; E ) sin ( π a ) R a Z a / π \sin(\pi a)R_aZ_a/\pi sin ( π a ) R a Z a / π Θ \Theta Θ [ 0 , T ] [0,T] [ 0 , T ] R + \mathbb{R}_+ R +
Finally, split (29) at time t t t
Θ t + h = S h Θ t + ∑ k = 1 ∞ ∫ t t + h S t + h − u f ( u ) G b e k d β k ( u ) . \Theta_{t+h}=S_h\Theta_t+\sum_{k=1}^{\infty}\int_t^{t+h}S_{t+h-u}\sqrt{f(u)}G_be_k\,\mathrm{d}\beta_k(u). Θ t + h = S h Θ t + k = 1 ∑ ∞ ∫ t t + h S t + h − u f ( u ) G b e k d β k ( u ) . Applying δ r \delta_r δ r 0 ≤ s ≤ t 0\le s\le t 0 ≤ s ≤ t
Ξ s , t : = ∑ k = 1 ∞ ∫ s t S t − u f ( u ) G b e k d β k ( u ) . \Xi_{s,t}:=\sum_{k=1}^{\infty}\int_s^t S_{t-u}\sqrt{f(u)}G_be_k\,\mathrm{d}\beta_k(u). Ξ s , t := k = 1 ∑ ∞ ∫ s t S t − u f ( u ) G b e k d β k ( u ) . Then Ξ s , t \Xi_{s,t} Ξ s , t F s \mathcal{F}_s F s F : E → R F:E\to\mathbb{R} F : E → R
( P s , t F ) ( θ ) : = E F ( S t − s θ + Ξ s , t ) . (P_{s,t}F)(\theta):=\mathbb{E}F(S_{t-s}\theta+\Xi_{s,t}). ( P s , t F ) ( θ ) := E F ( S t − s θ + Ξ s , t ) . The preceding decomposition gives E [ F ( Θ t ) ∣ F s ] = ( P s , t F ) ( Θ s ) \mathbb{E}[F(\Theta_t)\mid\mathcal{F}_s]=(P_{s,t}F)(\Theta_s) E [ F ( Θ t ) ∣ F s ] = ( P s , t F ) ( Θ s ) s ≤ r ≤ t s\le r\le t s ≤ r ≤ t
Ξ s , t = S t − r Ξ s , r + Ξ r , t , \Xi_{s,t}=S_{t-r}\Xi_{s,r}+\Xi_{r,t}, Ξ s , t = S t − r Ξ s , r + Ξ r , t , with Ξ r , t \Xi_{r,t} Ξ r , t F r \mathcal{F}_r F r P s , t = P s , r P r , t P_{s,t}=P_{s,r}P_{r,t} P s , t = P s , r P r , t F ∈ C b ( E ) F\in C_b(E) F ∈ C b ( E ) θ n → θ \theta_n\to\theta θ n → θ E E E F ( S t − s θ n + Ξ s , t ) → F ( S t − s θ + Ξ s , t ) F(S_{t-s}\theta_n+\Xi_{s,t})\to F(S_{t-s}\theta+\Xi_{s,t}) F ( S t − s θ n + Ξ s , t ) → F ( S t − s θ + Ξ s , t ) P s , t F ∈ C b ( E ) P_{s,t}F\in C_b(E) P s , t F ∈ C b ( E ) F : R + × E → R F:\mathbb{R}_+\times E\to\mathbb{R} F : R + × E → R
( P h F ) ( s , θ ) : = E F ( s + h , S h θ + Ξ s , s + h ) . (P_hF)(s,\theta):=\mathbb{E}F(s+h,S_h\theta+\Xi_{s,s+h}). ( P h F ) ( s , θ ) := E F ( s + h , S h θ + Ξ s , s + h ) . The deterministic kernels are measurable in their time parameters, and the convolution identity above gives P h + k = P h P k P_{h+k}=P_hP_k P h + k = P h P k R + × E \mathbb{R}_+\times E R + × E □ \square □
Remark 2.8 The forward curve has the conditional-mean interpretation
Θ t ( r ) = E [ ζ t + r , f , b ∣ F t ] , t , r ≥ 0. (34) \Theta_t(r)=\mathbb{E}[\zeta_{t+r,f,b}\mid\mathcal{F}_t],\qquad t,r\geq0. \tag*{(34)} Θ t ( r ) = E [ ζ t + r , f , b ∣ F t ] , t , r ≥ 0. (34) Indeed, split the stochastic integral for ζ t + r , f , b \zeta_{t+r,f,b} ζ t + r , f , b t t t F t \mathcal{F}_t F t h ≥ 0 h\geq0 h ≥ 0
ζ t + h , f , b − ζ t , f , b = Θ t ( h ) − Θ t ( 0 ) + ∑ k = 1 ∞ ∫ t t + h f ( u ) ⟨ ϕ t + h − u , e k ⟩ U b d β k ( u ) . (35) \zeta_{t+h,f,b}-\zeta_{t,f,b}=\Theta_t(h)-\Theta_t(0)+\sum_{k=1}^{\infty}\int_t^{t+h}\sqrt{f(u)}\langle\phi_{t+h-u},e_k\rangle_{U_b}\,\mathrm{d}\beta_k(u). \tag*{(35)} ζ t + h , f , b − ζ t , f , b = Θ t ( h ) − Θ t ( 0 ) + k = 1 ∑ ∞ ∫ t t + h f ( u ) ⟨ ϕ t + h − u , e k ⟩ U b d β k ( u ) . (35) Thus the present curve determines the contribution of past noise to every future increment. The remaining Gaussian process uses only fresh Brownian increments; its covariance at future lags h , k ≥ 0 h,k\geq0 h , k ≥ 0
C t ( h , k ) = ∫ t t + min ( h , k ) f ( u ) Q b ( t + h − u , t + k − u ) d u . C_t(h,k)=\int_t^{t+\min(h,k)}f(u)Q_b(t+h-u,t+k-u)\,\mathrm{d}u. C t ( h , k ) = ∫ t t + m i n ( h , k ) f ( u ) Q b ( t + h − u , t + k − u ) d u . The full curve therefore determines the conditional law of future increments. The boundary value alone need not do so.
Although δ 0 G b = 0 \delta_0G_b=0 δ 0 G b = 0 δ 0 S q G b h = ⟨ ϕ q , h ⟩ U b \delta_0S_qG_bh=\langle\phi_q,h\rangle_{U_b} δ 0 S q G b h = ⟨ ϕ q , h ⟩ U b q > 0 q>0 q > 0
E ∣ Θ t ( 0 ) ∣ 2 = κ b ∫ 0 t f ( u ) ( t − u ) b d u , \mathbb{E}|\Theta_t(0)|^2=\kappa_b\int_0^t f(u)(t-u)^b\,\mathrm{d}u, E ∣ Θ t ( 0 ) ∣ 2 = κ b ∫ 0 t f ( u ) ( t − u ) b d u , which is positive exactly when f > 0 f>0 f > 0 ( 0 , t ) (0,t) ( 0 , t ) δ 0 \delta_0 δ 0 δ 0 A Θ t \delta_0A\Theta_t δ 0 A Θ t
All these conditional statements use the given Brownian filtration; no identification with the natural filtration of the scalar boundary is assumed. At b = 1 b=1 b = 1 U 1 = L 2 ( ( 0 , ∞ ) , d λ ) U_1=L^2((0,\infty),\mathrm{d}\lambda) U 1 = L 2 (( 0 , ∞ ) , d λ ) 0 < γ < 1 0<\gamma<1 0 < γ < 1
Young and finite-variation equations
We first establish the path regularity needed to integrate against the boundary process. We then solve the equation from every lifted state and prove that the pair ( X , Θ ) (X,\Theta) ( X , Θ ) C b C_b C b f f f
The regularity dichotomy
Proposition 3.1 Let T > 0 T>0 T > 0 0 < b < 1 0<b<1 0 < b < 1 f ∈ L ∞ ( 0 , T ) f\in L^\infty(0,T) f ∈ L ∞ ( 0 , T ) 0 ≤ s < t ≤ T 0\le s<t\le T 0 ≤ s < t ≤ T
E ∣ ζ t , f , b − ζ s , f , b ∣ 2 ≤ ∥ f ∥ ∞ , T 2 1 − b 2 ∣ t − s ∣ 1 + b . (36) \mathbb{E}\left|\zeta_{t,f,b}-\zeta_{s,f,b}\right|^2\le\frac{\|f\|_{\infty,T}^2}{1-b^2}|t-s|^{1+b}. \tag*{(36)} E ∣ ζ t , f , b − ζ s , f , b ∣ 2 ≤ 1 − b 2 ∥ f ∥ ∞ , T 2 ∣ t − s ∣ 1 + b . (36) Consequently, ζ f , b \zeta_{f,b} ζ f , b
0 < η < H b ∘ . (37) 0<\eta<H_b^\circ. \tag*{(37)} 0 < η < H b ∘ . (37) In particular, the paths can be used as Young integrators: for 1 / 2 < η < H b ∘ 1/2<\eta<H_b^\circ 1/2 < η < H b ∘ q q q η + 1 / q > 1 \eta+1/q>1 η + 1/ q > 1
Proof. Let h = t − s h=t-s h = t − s
E ∣ ζ t , f , b − ζ s , f , b ∣ 2 = ∫ 0 s f ( u ) ∥ ϕ t − u − ϕ s − u ∥ U b 2 d u + ∫ s t f ( u ) ∥ ϕ t − u ∥ U b 2 d u . \mathbb{E}\left|\zeta_{t,f,b}-\zeta_{s,f,b}\right|^2=\int_0^s f(u)\|\phi_{t-u}-\phi_{s-u}\|_{U_b}^2\,du+\int_s^t f(u)\|\phi_{t-u}\|_{U_b}^2\,du. E ∣ ζ t , f , b − ζ s , f , b ∣ 2 = ∫ 0 s f ( u ) ∥ ϕ t − u − ϕ s − u ∥ U b 2 d u + ∫ s t f ( u ) ∥ ϕ t − u ∥ U b 2 d u . Since ϕ r + h ( λ ) − ϕ r ( λ ) = b e − λ r ( 1 − e − λ h ) / λ \phi_{r+h}(\lambda)-\phi_r(\lambda)=\sqrt{b}e^{-\lambda r}(1-e^{-\lambda h})/\lambda ϕ r + h ( λ ) − ϕ r ( λ ) = b e − λ r ( 1 − e − λh ) / λ r = s − u r=s-u r = s − u ( 0 , ∞ ) (0,\infty) ( 0 , ∞ )
∫ 0 s ∥ ϕ r + h − ϕ r ∥ U b 2 d r ≤ b 2 Γ ( 2 − b ) ∫ 0 ∞ ( 1 − e − λ h ) 2 λ − 2 − b d λ = 2 b 1 − b 2 h 1 + b . \int_0^s\|\phi_{r+h}-\phi_r\|_{U_b}^2\,dr\le\frac{b}{2\Gamma(2-b)}\int_0^\infty(1-e^{-\lambda h})^2\lambda^{-2-b}\,d\lambda=\frac{2^b}{1-b^2}h^{1+b}. ∫ 0 s ∥ ϕ r + h − ϕ r ∥ U b 2 d r ≤ 2Γ ( 2 − b ) b ∫ 0 ∞ ( 1 − e − λh ) 2 λ − 2 − b d λ = 1 − b 2 2 b h 1 + b . For the last equality, rescale x = λ h x=\lambda h x = λh ∫ 0 ∞ ( 1 − e − x ) 2 x − 2 − b d x = 2 Γ ( 1 − b ) ( 2 b − 1 ) / ( b ( 1 + b ) ) \int_0^\infty(1-e^{-x})^2x^{-2-b}\,dx=2\Gamma(1-b)(2^b-1)/(b(1+b)) ∫ 0 ∞ ( 1 − e − x ) 2 x − 2 − b d x = 2Γ ( 1 − b ) ( 2 b − 1 ) / ( b ( 1 + b )) ∥ f ∥ ∞ , T κ b h 1 + b / ( 1 + b ) = ∥ f ∥ ∞ , T ( 2 − 2 b ) h 1 + b / ( 1 − b 2 ) \|f\|_{\infty,T}\kappa_bh^{1+b}/(1+b)=\|f\|_{\infty,T}(2-2^b)h^{1+b}/(1-b^2) ∥ f ∥ ∞ , T κ b h 1 + b / ( 1 + b ) = ∥ f ∥ ∞ , T ( 2 − 2 b ) h 1 + b / ( 1 − b 2 ) E ∣ ζ t − ζ s ∣ q ≤ C q , b , f , T ∣ t − s ∣ q H b ∘ \mathbb{E}|\zeta_t-\zeta_s|^q\le C_{q,b,f,T}|t-s|^{qH_b^\circ} E ∣ ζ t − ζ s ∣ q ≤ C q , b , f , T ∣ t − s ∣ q H b ∘ q ≥ 2 q\ge2 q ≥ 2 [11 Section 3.2] gives every exponent η < H b ∘ − 1 / q \eta<H_b^\circ-1/q η < H b ∘ − 1/ q q → ∞ q\to\infty q → ∞
Proposition 3.2 Let T > 0 T>0 T > 0 b = 1 b=1 b = 1 f ∈ L ∞ ( 0 , T ) f\in L^\infty(0,T) f ∈ L ∞ ( 0 , T ) 0 ≤ s < t ≤ T 0\le s<t\le T 0 ≤ s < t ≤ T h : = t − s h:=t-s h := t − s
E ∣ ζ t , f , 1 − ζ s , f , 1 ∣ 2 ≤ ∥ f ∥ ∞ , T 2 h 2 ( 3 log 2 + 1 2 log T h ) . (38) \mathbb{E}\left|\zeta_{t,f,1}-\zeta_{s,f,1}\right|^2\le\|f\|_{\infty,T}^2h^2\left(3\log2+\frac{1}{2}\log\frac{T}{h}\right). \tag*{(38)} E ∣ ζ t , f , 1 − ζ s , f , 1 ∣ 2 ≤ ∥ f ∥ ∞ , T 2 h 2 ( 3 log 2 + 2 1 log h T ) . (38) Consequently, ζ f , 1 \zeta_{f,1} ζ f , 1 p p p p > 1 p>1 p > 1
Proof. At b = 1 b=1 b = 1 U 1 = L 2 ( ( 0 , ∞ ) , d λ ) U_1=L^2((0,\infty),d\lambda) U 1 = L 2 (( 0 , ∞ ) , d λ ) ϕ v ′ ( λ ) = e − λ v \phi'_v(\lambda)=e^{-\lambda v} ϕ v ′ ( λ ) = e − λ v r , h ≥ 0 r,h\ge0 r , h ≥ 0
D 1 ( r , h ) : = ∥ ϕ r + h − ϕ r ∥ U 1 2 = ∫ r r + h ∫ r r + h d v d w v + w . (39) D_1(r,h):=\|\phi_{r+h}-\phi_r\|_{U_1}^2=\int_r^{r+h}\int_r^{r+h}\frac{dv\,dw}{v+w}. \tag*{(39)} D 1 ( r , h ) := ∥ ϕ r + h − ϕ r ∥ U 1 2 = ∫ r r + h ∫ r r + h v + w d v d w . (39) Hence D 1 ( r , h ) ≤ D 1 ( 0 , h ) = 2 h log 2 D_1(r,h)\le D_1(0,h)=2h\log2 D 1 ( r , h ) ≤ D 1 ( 0 , h ) = 2 h log 2 D 1 ( r , h ) ≤ h 2 / ( 2 r ) D_1(r,h)\le h^2/(2r) D 1 ( r , h ) ≤ h 2 / ( 2 r ) r > 0 r>0 r > 0
E ∣ ζ t , f , 1 − ζ s , f , 1 ∣ 2 = ∫ 0 s f ( u ) D 1 ( s − u , h ) d u + 2 log 2 ∫ s t f ( u ) ( t − u ) d u . \mathbb{E}\left|\zeta_{t,f,1}-\zeta_{s,f,1}\right|^2=\int_0^s f(u)D_1(s-u,h)\,du+2\log2\int_s^t f(u)(t-u)\,du. E ∣ ζ t , f , 1 − ζ s , f , 1 ∣ 2 = ∫ 0 s f ( u ) D 1 ( s − u , h ) d u + 2 log 2 ∫ s t f ( u ) ( t − u ) d u . Write log + x : = max { log x , 0 } \log_+x:=\max\{\log x,0\} log + x := max { log x , 0 } x > 0 x>0 x > 0 log + 0 : = 0 \log_+0:=0 log + 0 := 0 s − u = h s-u=h s − u = h ∥ f ∥ ∞ , T ( 2 h 2 log 2 + h 2 log + ( s / h ) ) \|f\|_{\infty,T}(2h^2\log2+h^2\log_+(s/h)) ∥ f ∥ ∞ , T ( 2 h 2 log 2 + h 2 log + ( s / h )) ∥ f ∥ ∞ , T h 2 log 2 \|f\|_{\infty,T}h^2\log2 ∥ f ∥ ∞ , T h 2 log 2 log + ( s / h ) ≤ log ( T / h ) \log_+(s/h)\le\log(T/h) log + ( s / h ) ≤ log ( T / h )
For every H < 1 H<1 H < 1 h 2 log ( e T / h ) ≤ C H , T h 2 H h^2\log(eT/h)\le C_{H,T}h^{2H} h 2 log ( e T / h ) ≤ C H , T h 2 H ρ \rho ρ ρ < 1 \rho<1 ρ < 1 ρ m ↑ 1 \rho_m\uparrow1 ρ m ↑ 1 p > 1 p>1 p > 1 ρ > 1 / p \rho>1/p ρ > 1/ p p p p π \pi π ρ > 1 / 2 \rho>1/2 ρ > 1/2
∑ [ u , v ] ∈ π ∣ ζ v − ζ u ∣ 2 ≤ [ ζ ] ρ ; [ 0 , T ] 2 T ∣ π ∣ 2 ρ − 1 → 0. \sum_{[u,v]\in\pi}|\zeta_v-\zeta_u|^2\le[\zeta]^2_{\rho;[0,T]}T|\pi|^{2\rho-1}\to0. [ u , v ] ∈ π ∑ ∣ ζ v − ζ u ∣ 2 ≤ [ ζ ] ρ ; [ 0 , T ] 2 T ∣ π ∣ 2 ρ − 1 → 0. Proposition 3.3. Let 0 < b < 2 0<b<2 0 < b < 2 0 < γ < 1 0<\gamma<1 0 < γ < 1 b + γ > 1 b+\gamma>1 b + γ > 1 f ∈ L l o c 1 ( R + ) f\in L^1_{\mathrm{loc}}(\mathbb{R}_+) f ∈ L loc 1 ( R + ) Θ \Theta Θ Theorem 2.7 . For a fixed t > 0 t>0 t > 0 ∂ r Θ t ( 0 ) \partial_r\Theta_t(0) ∂ r Θ t ( 0 ) h − 1 ( Θ t ( h ) − Θ t ( 0 ) ) h^{-1}(\Theta_t(h)-\Theta_t(0)) h − 1 ( Θ t ( h ) − Θ t ( 0 )) L 2 ( Ω ) L^2(\Omega) L 2 ( Ω ) h ↓ 0 h\downarrow0 h ↓ 0
I f , b ( t ) : = ∫ 0 t f ( u ) ( t − u ) b − 2 d u < ∞ . (40) I_{f,b}(t):=\int_0^t f(u)(t-u)^{b-2}\,\mathrm{d}u<\infty. \tag*{(40)} I f , b ( t ) := ∫ 0 t f ( u ) ( t − u ) b − 2 d u < ∞. (40) When it exists,
∂ r Θ t ( 0 ) = ∑ k = 1 ∞ ∫ 0 t f ( u ) ⟨ ϕ t − u ′ , e k ⟩ U b d β k ( u ) , E ∣ ∂ r Θ t ( 0 ) ∣ 2 = b b − 2 I f , b ( t ) . \partial_r\Theta_t(0)=\sum_{k=1}^{\infty}\int_0^t\sqrt{f(u)}\langle\phi'_{t-u},e_k\rangle_{U_b}\,\mathrm{d}\beta_k(u),\qquad\mathbb{E}|\partial_r\Theta_t(0)|^2=b^{b-2}I_{f,b}(t). ∂ r Θ t ( 0 ) = k = 1 ∑ ∞ ∫ 0 t f ( u ) ⟨ ϕ t − u ′ , e k ⟩ U b d β k ( u ) , E ∣ ∂ r Θ t ( 0 ) ∣ 2 = b b − 2 I f , b ( t ) . If 0 < b ≤ 1 0<b\le1 0 < b ≤ 1 f ≥ c > 0 f\ge c>0 f ≥ c > 0 ( t − ε , t ) (t-\varepsilon,t) ( t − ε , t ) 0 < ε ≤ t 0<\varepsilon\le t 0 < ε ≤ t I f , b ( t ) = ∞ I_{f,b}(t)=\infty I f , b ( t ) = ∞
Proof. For h , q > 0 h,q>0 h , q > 0 h − 1 ( ϕ q + h − ϕ q ) ( λ ) = b e − λ q ( 1 − e − λ h ) / ( λ h ) h^{-1}(\phi_{q+h}-\phi_q)(\lambda)=\sqrt{b}e^{-\lambda q}(1-e^{-\lambda h})/(\lambda h) h − 1 ( ϕ q + h − ϕ q ) ( λ ) = b e − λ q ( 1 − e − λh ) / ( λh ) ϕ q ′ ( λ ) \phi'_q(\lambda) ϕ q ′ ( λ ) h ↓ 0 h\downarrow0 h ↓ 0
lim h ↓ 0 E ∣ Θ t ( h ) − Θ t ( 0 ) h ∣ 2 = ∫ 0 t f ( u ) ∥ ϕ t − u ′ ∥ U b 2 d u = b b − 2 I f , b ( t ) , \lim_{h\downarrow0}\mathbb{E}\left|\frac{\Theta_t(h)-\Theta_t(0)}{h}\right|^2=\int_0^t f(u)\|\phi'_{t-u}\|_{U_b}^2\,\mathrm{d}u=b^{b-2}I_{f,b}(t), h ↓ 0 lim E h Θ t ( h ) − Θ t ( 0 ) 2 = ∫ 0 t f ( u ) ∥ ϕ t − u ′ ∥ U b 2 d u = b b − 2 I f , b ( t ) , with value + ∞ +\infty + ∞ L 2 L^2 L 2 f f f I f , b ( t ) ≥ c ∫ 0 ε q b − 2 d q = ∞ I_{f,b}(t)\ge c\int_0^\varepsilon q^{b-2}\,\mathrm{d}q=\infty I f , b ( t ) ≥ c ∫ 0 ε q b − 2 d q = ∞ 0 < b ≤ 1 0<b\le1 0 < b ≤ 1
:::
Proposition 3.3 concerns differentiation in L 2 ( Ω ) L^2(\Omega) L 2 ( Ω ) f ∈ L l o c ∞ ( R + ) f\in L^\infty_{\mathrm{loc}}(\mathbb{R}_+) f ∈ L loc ∞ ( R + ) 1 < b < 2 1<b<2 1 < b < 2
Proposition 3.4. Let 1 < b < 2 1<b<2 1 < b < 2 0 < γ < 1 0<\gamma<1 0 < γ < 1 f ∈ L l o c ∞ ( R + ) f\in L^\infty_{\mathrm{loc}}(\mathbb{R}_+) f ∈ L loc ∞ ( R + ) Θ \Theta Θ Theorem 2.7 . For every t > 0 t>0 t > 0 r ↦ Θ t ( r ) r\mapsto\Theta_t(r) r ↦ Θ t ( r ) r = 0 r=0 r = 0 L 2 ( Ω ) L^2(\Omega) L 2 ( Ω )
V t : = ∂ r Θ t ( 0 ) = ∑ k = 1 ∞ ∫ 0 t f ( u ) ⟨ ϕ t − u ′ , e k ⟩ U b d β k ( u ) . (41) V_t:=\partial_r\Theta_t(0)=\sum_{k=1}^{\infty}\int_0^t\sqrt{f(u)}\langle\phi'_{t-u},e_k\rangle_{U_b}\,\mathrm{d}\beta_k(u). \tag*{(41)} V t := ∂ r Θ t ( 0 ) = k = 1 ∑ ∞ ∫ 0 t f ( u ) ⟨ ϕ t − u ′ , e k ⟩ U b d β k ( u ) . (41) Set V 0 : = 0 V_0:=0 V 0 := 0
E [ V s V t ] = b ∫ 0 s ∧ t f ( u ) ( s + t − 2 u ) b − 2 d u . (42) \mathbb{E}[V_sV_t]=b\int_0^{s\wedge t}f(u)(s+t-2u)^{b-2}\,\mathrm{d}u. \tag*{(42)} E [ V s V t ] = b ∫ 0 s ∧ t f ( u ) ( s + t − 2 u ) b − 2 d u . (42) Moreover,
ζ t , f , b = ∫ 0 t V s d s for all t ≥ 0 on one event of probability one. (43) \zeta_{t,f,b}=\int_0^tV_s\,\mathrm{d}s\qquad\text{for all }t\ge0\text{ on one event of probability one.} \tag*{(43)} ζ t , f , b = ∫ 0 t V s d s for all t ≥ 0 on one event of probability one. (43) Hence ζ f , b \zeta_{f,b} ζ f , b
Proof. For q , h > 0 q,h>0 q , h > 0
∥ ϕ q + h − ϕ q h ∥ U b 2 = ∥ ∫ 0 1 ϕ q + v h ′ d v ∥ U b 2 ≤ ∫ 0 1 ∥ ϕ q + v h ′ ∥ U b 2 d v ≤ b ( 2 q ) b − 2 , \left\Vert\frac{\phi_{q+h}-\phi_q}{h}\right\Vert_{U_b}^{2}
=\left\Vert\int_0^1\phi'_{q+vh}\,dv\right\Vert_{U_b}^{2}
\leq\int_0^1\left\Vert\phi'_{q+vh}\right\Vert_{U_b}^{2}\,dv
\leq b(2q)^{b-2}, h ϕ q + h − ϕ q U b 2 = ∫ 0 1 ϕ q + v h ′ d v U b 2 ≤ ∫ 0 1 ϕ q + v h ′ U b 2 d v ≤ b ( 2 q ) b − 2 , because b − 2 < 0 b-2<0 b − 2 < 0 q = 0 q=0 q = 0 b > 1 b>1 b > 1 ⟨ ϕ r ′ , ϕ q ′ ⟩ U b = b ( r + q ) b − 2 \langle\phi'_r,\phi'_q\rangle_{U_b}=b(r+q)^{b-2} ⟨ ϕ r ′ , ϕ q ′ ⟩ U b = b ( r + q ) b − 2
E ∫ 0 T ∣ V s ∣ 2 d s = b 2 b − 2 ∫ 0 T ∫ 0 s f ( u ) ( s − u ) b − 2 d u d s ≤ 2 b − 2 ∥ f ∥ ∞ , T T b b − 1 < ∞ . \mathbb{E}\int_0^T|V_s|^2\,ds
=b2^{b-2}\int_0^T\int_0^s f(u)(s-u)^{b-2}\,du\,ds
\leq\frac{2^{b-2}\|f\|_{\infty,T}T^b}{b-1}<\infty. E ∫ 0 T ∣ V s ∣ 2 d s = b 2 b − 2 ∫ 0 T ∫ 0 s f ( u ) ( s − u ) b − 2 d u d s ≤ b − 1 2 b − 2 ∥ f ∥ ∞ , T T b < ∞. The map s ↦ 1 { u < s } f ( u ) ϕ s − u ′ s\mapsto\mathbf{1}_{\{u<s\}}\sqrt{f(u)}\phi'_{s-u} s ↦ 1 { u < s } f ( u ) ϕ s − u ′ L 2 ( 0 , T ; U b ) L^2(0,T;U_b) L 2 ( 0 , T ; U b ) V V V V ∈ L 2 ( 0 , T ) V\in L^2(0,T) V ∈ L 2 ( 0 , T ) [11 Section 4.5] holds, since
∫ 0 T ( ∫ 0 s f ( u ) ∥ ϕ s − u ′ ∥ U b 2 d u ) 1 / 2 d s ≤ T 1 / 2 ( E ∫ 0 T ∣ V s ∣ 2 d s ) 1 / 2 < ∞ . \int_0^T\left(\int_0^s f(u)\|\phi'_{s-u}\|_{U_b}^2\,du\right)^{1/2}ds
\leq T^{1/2}\left(\mathbb{E}\int_0^T|V_s|^2\,ds\right)^{1/2}<\infty. ∫ 0 T ( ∫ 0 s f ( u ) ∥ ϕ s − u ′ ∥ U b 2 d u ) 1/2 d s ≤ T 1/2 ( E ∫ 0 T ∣ V s ∣ 2 d s ) 1/2 < ∞. Using ϕ q = ∫ 0 q ϕ r ′ d r \phi_q=\int_0^q\phi'_r\,dr ϕ q = ∫ 0 q ϕ r ′ d r t ≤ T t\leq T t ≤ T
∫ 0 t V s d s = ∑ k = 1 ∞ ∫ 0 t f ( u ) ( ∫ u t ⟨ ϕ s − u ′ , e k ⟩ U b d s ) d β k ( u ) = ζ t , f , b . \int_0^t V_s\,ds
=\sum_{k=1}^{\infty}\int_0^t\sqrt{f(u)}
\left(\int_u^t\langle\phi'_{s-u},e_k\rangle_{U_b}\,ds\right)d\beta_k(u)
=\zeta_{t,f,b}. ∫ 0 t V s d s = k = 1 ∑ ∞ ∫ 0 t f ( u ) ( ∫ u t ⟨ ϕ s − u ′ , e k ⟩ U b d s ) d β k ( u ) = ζ t , f , b . For each fixed t t t V ∈ L 2 ( 0 , T ) V\in L^2(0,T) V ∈ L 2 ( 0 , T ) t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ] T T T
Coefficient regularity and pathwise well-posedness
For h : R → R h:\mathbb{R}\to\mathbb{R} h : R → R ∥ h ∥ ∞ : = sup x ∈ R ∣ h ( x ) ∣ \|h\|_\infty:=\sup_{x\in\mathbb{R}}|h(x)| ∥ h ∥ ∞ := sup x ∈ R ∣ h ( x ) ∣ 0 < α ≤ 1 0<\alpha\leq1 0 < α ≤ 1
[ h ] α : = sup x ≠ y ∣ h ( x ) − h ( y ) ∣ ∣ x − y ∣ α . [h]_\alpha:=\sup_{x\ne y}\frac{|h(x)-h(y)|}{|x-y|^\alpha}. [ h ] α := x = y sup ∣ x − y ∣ α ∣ h ( x ) − h ( y ) ∣ . At α = 1 \alpha=1 α = 1
Assumption 3.5 The functions a , σ : R × R + → R a,\sigma:\mathbb{R}\times\mathbb{R}_+\to\mathbb{R} a , σ : R × R + → R T > 0 T>0 T > 0 0 ≤ L T , K T < ∞ 0\leq L_T,K_T<\infty 0 ≤ L T , K T < ∞
∣ a ( x , t ) − a ( y , t ) ∣ + ∣ σ ( x , t ) − σ ( y , t ) ∣ ≤ L T ∣ x − y ∣ , x , y ∈ R , t ∈ [ 0 , T ] , |a(x,t)-a(y,t)|+|\sigma(x,t)-\sigma(y,t)|\leq L_T|x-y|,
\qquad x,y\in\mathbb{R},\quad t\in[0,T], ∣ a ( x , t ) − a ( y , t ) ∣ + ∣ σ ( x , t ) − σ ( y , t ) ∣ ≤ L T ∣ x − y ∣ , x , y ∈ R , t ∈ [ 0 , T ] , ∣ a ( 0 , t ) ∣ + ∣ σ ( 0 , t ) ∣ ≤ K T . |a(0,t)|+|\sigma(0,t)|\leq K_T. ∣ a ( 0 , t ) ∣ + ∣ σ ( 0 , t ) ∣ ≤ K T . Continuity gives the bound at the origin. Together the two bounds imply
∣ a ( x , t ) ∣ + ∣ σ ( x , t ) ∣ ≤ K T + L T ∣ x ∣ . |a(x,t)|+|\sigma(x,t)|\leq K_T+L_T|x|. ∣ a ( x , t ) ∣ + ∣ σ ( x , t ) ∣ ≤ K T + L T ∣ x ∣. When 0 < b ≤ 1 0<b\leq1 0 < b ≤ 1 α ∈ ( 0 , 1 ] \alpha\in(0,1] α ∈ ( 0 , 1 ]
α > ( H b 0 ) − 1 − 1 , (44) \alpha>(H_b^0)^{-1}-1, \tag*{(44)} α > ( H b 0 ) − 1 − 1 , (44) and σ \sigma σ x x x
sup 0 ≤ t ≤ T ( ∥ ∂ x σ ( ⋅ , t ) ∥ ∞ + [ ∂ x σ ( ⋅ , t ) ] α ) ≤ K T . (45) \sup_{0\leq t\leq T}\left(\|\partial_x\sigma(\cdot,t)\|_\infty+[\partial_x\sigma(\cdot,t)]_\alpha\right)\leq K_T. \tag*{(45)} 0 ≤ t ≤ T sup ( ∥ ∂ x σ ( ⋅ , t ) ∥ ∞ + [ ∂ x σ ( ⋅ , t ) ] α ) ≤ K T . (45) sup x ∈ R ∣ σ ( x , t ) − σ ( x , s ) ∣ ≤ K T ∣ t − s ∣ , s , t ∈ [ 0 , T ] . (46) \sup_{x\in\mathbb{R}} |\sigma(x,t)-\sigma(x,s)| \le K_T |t-s|,\qquad s,t\in[0,T]. \tag*{(46)} x ∈ R sup ∣ σ ( x , t ) − σ ( x , s ) ∣ ≤ K T ∣ t − s ∣ , s , t ∈ [ 0 , T ] . (46) The constants may be enlarged to satisfy all these inequalities. For 1 < b < 2 1<b<2 1 < b < 2
The time regularity of σ \sigma σ ∂ x σ \partial_x\sigma ∂ x σ
Lemma 3.6 Under (45)–(46), put β ∗ = α / ( 1 + α ) \beta_*=\alpha/(1+\alpha) β ∗ = α / ( 1 + α ) T > 0 T>0 T > 0
sup x ∈ R ( ∣ σ ( x , t ) − σ ( x , s ) ∣ + ∣ ∂ x σ ( x , t ) − ∂ x σ ( x , s ) ∣ ) ≤ C T ∣ t − s ∣ β ∗ , s , t ∈ [ 0 , T ] . (47) \sup_{x\in\mathbb{R}}\left(|\sigma(x,t)-\sigma(x,s)|+|\partial_x\sigma(x,t)-\partial_x\sigma(x,s)|\right)\le C_T|t-s|^{\beta_*},\qquad s,t\in[0,T]. \tag*{(47)} x ∈ R sup ( ∣ σ ( x , t ) − σ ( x , s ) ∣ + ∣ ∂ x σ ( x , t ) − ∂ x σ ( x , s ) ∣ ) ≤ C T ∣ t − s ∣ β ∗ , s , t ∈ [ 0 , T ] . (47) Proof Fix s ≠ t s\ne t s = t [ 0 , T ] [0,T] [ 0 , T ] d : = ∣ t − s ∣ d:=|t-s| d := ∣ t − s ∣ g : = σ ( ⋅ , t ) − σ ( ⋅ , s ) g:=\sigma(\cdot,t)-\sigma(\cdot,s) g := σ ( ⋅ , t ) − σ ( ⋅ , s ) ∥ g ∥ ∞ ≤ K T d \|g\|_\infty\le K_Td ∥ g ∥ ∞ ≤ K T d [ g ′ ] α ≤ 2 K T [g']_\alpha\le2K_T [ g ′ ] α ≤ 2 K T x ∈ R x\in\mathbb{R} x ∈ R h > 0 h>0 h > 0
g ′ ( x ) = g ( x + h ) − g ( x ) h − 1 h ∫ 0 h ( g ′ ( x + r ) − g ′ ( x ) ) d r . g'(x)=\frac{g(x+h)-g(x)}{h}-\frac{1}{h}\int_0^h\bigl(g'(x+r)-g'(x)\bigr)\,dr. g ′ ( x ) = h g ( x + h ) − g ( x ) − h 1 ∫ 0 h ( g ′ ( x + r ) − g ′ ( x ) ) d r . Taking absolute values gives
∥ g ′ ∥ ∞ ≤ 2 K T d h + 2 K T 1 + α h α . \|g'\|_\infty\le\frac{2K_Td}{h}+\frac{2K_T}{1+\alpha}h^\alpha. ∥ g ′ ∥ ∞ ≤ h 2 K T d + 1 + α 2 K T h α . The choice h = d 1 / ( 1 + α ) h=d^{1/(1+\alpha)} h = d 1/ ( 1 + α ) ∥ g ′ ∥ ∞ ≤ 4 K T d β ∗ \|g'\|_\infty\le4K_Td^{\beta_*} ∥ g ′ ∥ ∞ ≤ 4 K T d β ∗ d ≤ T d\le T d ≤ T ∥ g ∥ ∞ ≤ K T T 1 − β ∗ d β ∗ \|g\|_\infty\le K_TT^{1-\beta_*}d^{\beta_*} ∥ g ∥ ∞ ≤ K T T 1 − β ∗ d β ∗ C T = K T ( 4 + T 1 − β ∗ ) C_T=K_T(4+T^{1-\beta_*}) C T = K T ( 4 + T 1 − β ∗ )
Path norms.
For p ≥ 1 p\ge1 p ≥ 1 C p -var ( [ 0 , T ] ) C^{p\text{-var}}([0,T]) C p -var ([ 0 , T ]) p p p
N p ( z ) : = ∣ z ( 0 ) ∣ + ∥ z ∥ p -var ; [ 0 , T ] ≥ ∥ z ∥ ∞ ; [ 0 , T ] . N_p(z):=|z(0)|+\|z\|_{p\text{-var};[0,T]}\ge\|z\|_{\infty;[0,T]}. N p ( z ) := ∣ z ( 0 ) ∣ + ∥ z ∥ p -var ; [ 0 , T ] ≥ ∥ z ∥ ∞ ; [ 0 , T ] . For a continuous path v v v B T ( v ) : = ∣ v ( 0 ) ∣ + TV [ 0 , T ] ( v ) B_T(v):=|v(0)|+\operatorname{TV}_{[0,T]}(v) B T ( v ) := ∣ v ( 0 ) ∣ + TV [ 0 , T ] ( v ) N p ( v ) ≤ B T ( v ) N_p(v)\le B_T(v) N p ( v ) ≤ B T ( v ) z z z η \eta η p η ≥ 1 p\eta\ge1 p η ≥ 1 ∥ z ∥ p -var ; [ 0 , T ] ≤ [ z ] η ; [ 0 , T ] T η \|z\|_{p\text{-var};[0,T]}\le[z]_{\eta;[0,T]}T^\eta ∥ z ∥ p -var ; [ 0 , T ] ≤ [ z ] η ; [ 0 , T ] T η ∑ i ( t i + 1 − t i ) p η ≤ T p η \sum_i(t_{i+1}-t_i)^{p\eta}\le T^{p\eta} ∑ i ( t i + 1 − t i ) p η ≤ T p η q q q 1 / p + 1 / q > 1 1/p+1/q>1 1/ p + 1/ q > 1 [9 Section 2.1, (2.5)–(2.7)] .
Proposition 3.7 Assume (45)–(46) and the continuity and global spatial Lipschitz hypotheses on the drift in Assumption 3.5. Let 1 < p < min ( 2 , 1 + α ) 1<p<\min(2,1+\alpha) 1 < p < min ( 2 , 1 + α ) z ∈ C p -var ( [ 0 , T ] ) z\in C^{p\text{-var}}([0,T]) z ∈ C p -var ([ 0 , T ]) v v v
x t = x 0 + ∫ 0 t a ( x r , r ) d r + ∫ 0 t σ ( x r , r ) d ( z r + v r ) (48) x_t=x_0+\int_0^t a(x_r,r)\,dr+\int_0^t \sigma(x_r,r)\,d(z_r+v_r) \tag*{(48)} x t = x 0 + ∫ 0 t a ( x r , r ) d r + ∫ 0 t σ ( x r , r ) d ( z r + v r ) (48) has a unique solution in C p -var ( [ 0 , T ] ) C^{p\text{-var}}([0,T]) C p -var ([ 0 , T ]) z z z v v v ( x 0 , z , v ) (x_0,z,v) ( x 0 , z , v ) z z z N p N_p N p v v v B T B_T B T
Proof. Set w = z + v w=z+v w = z + v N p ( v ) ≤ B T ( v ) N_p(v)\le B_T(v) N p ( v ) ≤ B T ( v ) p p p [9 Theorems 3.6 and 3.9] ; their diffusion coefficient g ( t , x ) g(t,x) g ( t , x ) σ ( x , t ) \sigma(x,t) σ ( x , t ) f ( t , x ) f(t,x) f ( t , x ) a ( x , t ) a(x,t) a ( x , t )
In their notation, take δ = α \delta=\alpha δ = α β = β ∗ = α / ( 1 + α ) \beta=\beta_*=\alpha/(1+\alpha) β = β ∗ = α / ( 1 + α ) ∂ x σ \partial_x\sigma ∂ x σ α \alpha α H 1 H_1 H 1 H 1 H_1 H 1 h ( s , t ) = C T 1 / β ∗ ( t − s ) h(s,t)=C_T^{1/\beta_*}(t-s) h ( s , t ) = C T 1/ β ∗ ( t − s ) r ∈ ( 1 / p , 1 ) r\in(1/p,1) r ∈ ( 1/ p , 1 ) α \alpha α α \alpha α r > 1 / 2 r>1/2 r > 1/2 p < 2 p<2 p < 2
∣ a ( x , t ) ∣ ≤ L T ∣ x ∣ + A T , A T : = sup t ≤ T ∣ a ( 0 , t ) ∣ < ∞ . |a(x,t)|\le L_T|x|+A_T,\qquad A_T:=\sup_{t\le T}|a(0,t)|<\infty. ∣ a ( x , t ) ∣ ≤ L T ∣ x ∣ + A T , A T := t ≤ T sup ∣ a ( 0 , t ) ∣ < ∞. The constant envelope A T A_T A T L 1 / ( 1 − r ) ( 0 , T ) L^{1/(1-r)}(0,T) L 1/ ( 1 − r ) ( 0 , T ) H 2 H_2 H 2 H 3 H_3 H 3
α > p − 1 , β ∗ > 1 − 1 p , α r > α p > 1 − 1 p . \alpha>p-1,\qquad\beta_*>1-\frac{1}{p},\qquad\alpha r>\frac{\alpha}{p}>1-\frac{1}{p}. α > p − 1 , β ∗ > 1 − p 1 , α r > p α > 1 − p 1 . To check the variation exponents used in those theorems, choose q 0 q_0 q 0 q q q
1 − 1 p < 1 q 0 < min { β ∗ , α r , α p , 1 2 } , 1 q 0 α < 1 q < 1 p . 1-\frac{1}{p}<\frac{1}{q_0}<\min\left\{\beta_*,\alpha r,\frac{\alpha}{p},\frac{1}{2}\right\},\qquad\frac{1}{q_0\alpha}<\frac{1}{q}<\frac{1}{p}. 1 − p 1 < q 0 1 < min { β ∗ , α r , p α , 2 1 } , q 0 α 1 < q 1 < p 1 . Both choices are possible: every entry in the first minimum exceeds 1 − 1 / p 1-1/p 1 − 1/ p 1 / ( q 0 α ) < 1 / p < r 1/(q_0\alpha)<1/p<r 1/ ( q 0 α ) < 1/ p < r [9 (3.2)–(3.3)] , after replacing their drift parameter by r r r q > p q>p q > p C q -var C^{q\text{-var}} C q -var max { p , 1 / r } = p \max\{p,1/r\}=p max { p , 1/ r } = p C p -var ⊆ C q -var C^{p\text{-var}}\subseteq C^{q\text{-var}} C p -var ⊆ C q -var C p -var C^{p\text{-var}} C p -var
Their Theorem 3.9 gives joint continuity of the evaluation map in time, initial value and driver. Suppose x 0 n → x 0 x_0^n\to x_0 x 0 n → x 0 N p ( z n − z ) → 0 N_p(z_n-z)\to0 N p ( z n − z ) → 0 B T ( v n − v ) → 0 B_T(v_n-v)\to0 B T ( v n − v ) → 0 N p ( ( z n + v n ) − ( z + v ) ) → 0 N_p((z_n+v_n)-(z+v))\to0 N p (( z n + v n ) − ( z + v )) → 0 [ 0 , T ] [0,T] [ 0 , T ] [ 0 , T ] [0,T] [ 0 , T ]
Finally, the coefficient bounds give
∣ σ ( x t , t ) − σ ( x s , s ) ∣ ≤ K T ∣ x t − x s ∣ + K T ∣ t − s ∣ . |\sigma(x_t,t)-\sigma(x_s,s)|\le K_T|x_t-x_s|+K_T|t-s|. ∣ σ ( x t , t ) − σ ( x s , s ) ∣ ≤ K T ∣ x t − x s ∣ + K T ∣ t − s ∣. Thus the integrand has finite p p p p < 2 p<2 p < 2 w w w z z z z z z v v v v v v v v v
Remark 3.8 For 0 < b < 1 0<b<1 0 < b < 1 (44) is α > ( 1 − b ) / ( 1 + b ) \alpha>(1-b)/(1+b) α > ( 1 − b ) / ( 1 + b ) b = 1 b=1 b = 1 α > 0 \alpha>0 α > 0 § 4 impose the stronger, separately stated Assumptions 4.4 and 4.11.
The nonlinear lifted equation
Throughout this subsection, fix 0 < b < 2 0<b<2 0 < b < 2 0 < γ < 1 0<\gamma<1 0 < γ < 1 b + γ > 1 b+\gamma>1 b + γ > 1 f ∈ L l o c ∞ ( R + ) f\in L^\infty_{\mathrm{loc}}(\mathbb{R}_+) f ∈ L loc ∞ ( R + )
Restarted states.
Fix s ≥ 0 s \ge0 s ≥ 0 ( x , θ ) ∈ R × E γ (x,\theta) \in\mathbb{R} \times E_\gamma ( x , θ ) ∈ R × E γ W W W U b U_b U b ( F r ) r ≥ 0 (\mathcal{F}_r)_{r\ge0} ( F r ) r ≥ 0 r r r
Θ r s , θ ( ρ ) : = θ ( r + ρ ) + ∑ k = 1 ∞ ∫ 0 r f ( s + u ) ⟨ ϕ r + ρ − u , e k ⟩ U b d β k ( u ) , r , ρ ≥ 0. (49) \Theta^{s,\theta}_r(\rho) := \theta(r+\rho) + \sum_{k=1}^{\infty}\int_0^r \sqrt{f(s+u)}\langle\phi_{r+\rho-u},e_k\rangle_{U_b}\,d\beta_k(u), \qquad r,\rho\ge0. \tag*{(49)} Θ r s , θ ( ρ ) := θ ( r + ρ ) + k = 1 ∑ ∞ ∫ 0 r f ( s + u ) ⟨ ϕ r + ρ − u , e k ⟩ U b d β k ( u ) , r , ρ ≥ 0. (49) Equivalently, Θ r s , θ = S r θ + Ξ r s \Theta^{s,\theta}_r=S_r\theta+\Xi^s_r Θ r s , θ = S r θ + Ξ r s E γ E_\gamma E γ Ξ s : = Θ s , 0 \Xi^s:=\Theta^{s,0} Ξ s := Θ s , 0 f ( s + ⋅ ) f(s+\cdot) f ( s + ⋅ ) Ξ s \Xi^s Ξ s E γ E_\gamma E γ Θ s , θ \Theta^{s,\theta} Θ s , θ δ 0 \delta_0 δ 0 E γ E_\gamma E γ
ζ r s , θ : = Θ r s , θ ( 0 ) = θ ( r ) + ζ r s , 0 , \zeta^{s,\theta}_r := \Theta^{s,\theta}_r(0)=\theta(r)+\zeta^{s,0}_r, ζ r s , θ := Θ r s , θ ( 0 ) = θ ( r ) + ζ r s , 0 , ζ r s , 0 : = Ξ r s ( 0 ) = ∑ k = 1 ∞ ∫ 0 r f ( s + u ) ⟨ ϕ r − u , e k ⟩ U b d β k ( u ) , \zeta^{s,0}_r:=\Xi^s_r(0)=\sum_{k=1}^{\infty}\int_0^r\sqrt{f(s+u)}\langle\phi_{r-u},e_k\rangle_{U_b}\,d\beta_k(u), ζ r s , 0 := Ξ r s ( 0 ) = k = 1 ∑ ∞ ∫ 0 r f ( s + u ) ⟨ ϕ r − u , e k ⟩ U b d β k ( u ) , is continuous in r r r ζ s , 0 \zeta^{s,0} ζ s , 0 f ( s + ⋅ ) f(s+\cdot) f ( s + ⋅ ) ϑ ∈ E γ \vartheta\in E_\gamma ϑ ∈ E γ T > 0 T>0 T > 0
B T ( ϑ ∣ [ 0 , T ] ) ≤ ( 1 + J γ ( T ) 1 / 2 ) ∥ ϑ ∥ γ , sup r ≤ T ∥ S r ϑ ∥ γ ≤ C γ , T ∥ ϑ ∥ γ . (50) B_T(\vartheta|_{[0,T]})\le(1+J_\gamma(T)^{1/2})\|\vartheta\|_\gamma,\qquad\sup_{r\le T}\|S_r\vartheta\|_\gamma\le C_{\gamma,T}\|\vartheta\|_\gamma. \tag*{(50)} B T ( ϑ ∣ [ 0 , T ] ) ≤ ( 1 + J γ ( T ) 1/2 ) ∥ ϑ ∥ γ , r ≤ T sup ∥ S r ϑ ∥ γ ≤ C γ , T ∥ ϑ ∥ γ . (50) In particular, restriction maps E γ E_\gamma E γ B T B_T B T θ ( ρ ) = ρ \theta(\rho)=\sqrt{\rho} θ ( ρ ) = ρ
∥ θ ∥ γ 2 = 1 4 ∫ 0 ∞ ρ γ − 1 ( 1 + ρ ) 2 d ρ < ∞ , 0 < γ < 1 , \|\theta\|_\gamma^2=\frac{1}{4}\int_0^\infty\frac{\rho^{\gamma-1}}{(1+\rho)^2}\,d\rho<\infty,\qquad0<\gamma<1, ∥ θ ∥ γ 2 = 4 1 ∫ 0 ∞ ( 1 + ρ ) 2 ρ γ − 1 d ρ < ∞ , 0 < γ < 1 , but is not η \eta η η > 1 / 2 \eta>1/2 η > 1/2 f ≡ 0 f\equiv0 f ≡ 0 r \sqrt{r} r E γ E_\gamma E γ
Remark 3.9 Fix r > 0 r>0 r > 0 θ \theta θ Ξ r s \Xi^s_r Ξ r s h − 1 ( Θ r s , θ ( h ) − Θ r s , θ ( 0 ) ) h^{-1}(\Theta^{s,\theta}_r(h)-\Theta^{s,\theta}_r(0)) h − 1 ( Θ r s , θ ( h ) − Θ r s , θ ( 0 )) h − 1 ( θ ( r + h ) − θ ( r ) ) h^{-1}(\theta(r+h)-\theta(r)) h − 1 ( θ ( r + h ) − θ ( r )) L 2 ( Ω ) L^2(\Omega) L 2 ( Ω ) h ↓ 0 h\downarrow0 h ↓ 0 θ + ′ ( r ) \theta'_+(r) θ + ′ ( r ) ∫ 0 r f ( s + u ) ( r − u ) b − 2 d u < ∞ \int_0^r f(s+u)(r-u)^{b-2}\,du<\infty ∫ 0 r f ( s + u ) ( r − u ) b − 2 d u < ∞ ∣ θ + ′ ( r ) ∣ 2 + b b 2 − b ∫ 0 r f ( s + u ) ( r − u ) b − 2 d u |\theta'_+(r)|^2+b b^{2-b}\int_0^r f(s+u)(r-u)^{b-2}\,du ∣ θ + ′ ( r ) ∣ 2 + b b 2 − b ∫ 0 r f ( s + u ) ( r − u ) b − 2 d u
Theorem 3.10. Let 0 < b < 2 0 < b < 2 0 < b < 2 0 < γ < 1 0 < \gamma< 1 0 < γ < 1 b + γ > 1 b + \gamma> 1 b + γ > 1 f ∈ L l o c ∞ ( R + ) f \in L^\infty_{\mathrm{loc}}(\mathbb{R}_{+}) f ∈ L loc ∞ ( R + ) s ≥ 0 s \ge0 s ≥ 0 ( x , θ ) ∈ R × E γ (x,\theta) \in\mathbb{R} \times E_\gamma ( x , θ ) ∈ R × E γ
X r s , x , θ = x + ∫ 0 r a ( X u s , x , θ , s + u ) d u + ∫ 0 r σ ( X u s , x , θ , s + u ) d ζ u s , θ X_{r}^{s,x,\theta}=x+\int_{0}^{r}a(X_{u}^{s,x,\theta},s+u)\,\mathrm{d}u+\int_{0}^{r}\sigma(X_{u}^{s,x,\theta},s+u)\,\mathrm{d}\zeta_{u}^{s,\theta} X r s , x , θ = x + ∫ 0 r a ( X u s , x , θ , s + u ) d u + ∫ 0 r σ ( X u s , x , θ , s + u ) d ζ u s , θ has the following pathwise solutions. For 1 < b < 2 1 < b < 2 1 < b < 2 0 < b ≤ 1 0 < b \le1 0 < b ≤ 1 ( H b ∘ ) − 1 < p < min ( 2 , 1 + α ) (H_b^\circ)^{-1} < p < \min(2,1+\alpha) ( H b ∘ ) − 1 < p < min ( 2 , 1 + α ) p p p θ \theta θ ζ s , 0 \zeta^{s,0} ζ s , 0 b = 1 b=1 b = 1 p ∈ ( 1 , 1 + α ) p \in(1,1+\alpha) p ∈ ( 1 , 1 + α )
The process X r s , x , θ : = ( X r s , x , θ , Θ s , θ ) X_{r}^{s,x,\theta}:=(X_{r}^{s,x,\theta},\Theta^{s,\theta}) X r s , x , θ := ( X r s , x , θ , Θ s , θ ) R × E γ \mathbb{R}\times E_\gamma R × E γ C b C_b C b
P u , t F ∈ C b ( R × E γ ) for every F ∈ C b ( R × E γ ) and 0 ≤ u ≤ t . P_{u,t}F\in C_b(\mathbb{R}\times E_\gamma)\quad\text{for every }F\in C_b(\mathbb{R}\times E_\gamma)\text{ and }0\le u\le t. P u , t F ∈ C b ( R × E γ ) for every F ∈ C b ( R × E γ ) and 0 ≤ u ≤ t . For s = 0 s=0 s = 0 θ = 0 \theta=0 θ = 0
Proof. Fix T > 0 T>0 T > 0 R × [ s , s + T ] \mathbb{R}\times[s,s+T] R × [ s , s + T ] L T L_T L T a a a σ \sigma σ
Finite-variation case.
Suppose 1 < b < 2 1<b<2 1 < b < 2 f ( s + ⋅ ) f(s+\cdot) f ( s + ⋅ ) ζ s , θ \zeta^{s,\theta} ζ s , θ μ ( d r ) : = d r + ∣ d ζ r s , θ ∣ \mu(\mathrm{d}r):=\mathrm{d}r+|\mathrm{d}\zeta_r^{s,\theta}| μ ( d r ) := d r + ∣ d ζ r s , θ ∣ [ 0 , T ] [0,T] [ 0 , T ]
Set x r ( 0 ) = x x_r^{(0)}=x x r ( 0 ) = x
x r ( m + 1 ) = x + ∫ 0 r a ( x u ( m ) , s + u ) d u + ∫ 0 r σ ( x u ( m ) , s + u ) d ζ u s , θ . x_r^{(m+1)}=x+\int_{0}^{r}a(x_u^{(m)},s+u)\,\mathrm{d}u+\int_{0}^{r}\sigma(x_u^{(m)},s+u)\,\mathrm{d}\zeta_u^{s,\theta}. x r ( m + 1 ) = x + ∫ 0 r a ( x u ( m ) , s + u ) d u + ∫ 0 r σ ( x u ( m ) , s + u ) d ζ u s , θ . They are continuous. Let A x : = sup u ≤ T ( ∣ a ( x , s + u ) ∣ + ∣ σ ( x , s + u ) ∣ ) < ∞ A_x:=\sup_{u\le T}(|a(x,s+u)|+|\sigma(x,s+u)|)<\infty A x := sup u ≤ T ( ∣ a ( x , s + u ) ∣ + ∣ σ ( x , s + u ) ∣ ) < ∞ A x μ ( [ 0 , r ] ) A_x\mu([0,r]) A x μ ([ 0 , r ]) μ \mu μ
sup u ≤ r ∣ x u ( m + 1 ) − x u ( m ) ∣ ≤ A x L T m μ ( [ 0 , r ] ) m + 1 ( m + 1 ) ! , m ≥ 0. \sup_{u\le r}|x_u^{(m+1)}-x_u^{(m)}|\le A_xL_T^m\frac{\mu([0,r])^{m+1}}{(m+1)!},\qquad m\ge0. u ≤ r sup ∣ x u ( m + 1 ) − x u ( m ) ∣ ≤ A x L T m ( m + 1 )! μ ([ 0 , r ] ) m + 1 , m ≥ 0. Indeed, each iteration integrates over an ordered simplex; its μ ⊗ ( m + 1 ) \mu^{\otimes(m+1)} μ ⊗ ( m + 1 ) μ ( [ 0 , r ] ) m + 1 / ( m + 1 ) ! \mu([0,r])^{m+1}/(m+1)! μ ([ 0 , r ] ) m + 1 / ( m + 1 )! [ 0 , T ] [0,T] [ 0 , T ]
The same ordered-simplex argument proves the Gronwall inequality for μ \mu μ D ≥ 0 D\ge0 D ≥ 0 D ( r ) ≤ a + L ∫ 0 r D d μ D(r)\le a+L\int_0^rD\,\mathrm{d}\mu D ( r ) ≤ a + L ∫ 0 r D d μ D ( r ) ≤ a exp ( L μ ( [ 0 , r ] ) ) D(r)\le a\exp(L\mu([0,r])) D ( r ) ≤ a exp ( Lμ ([ 0 , r ])) a = 0 a=0 a = 0 1 + ∣ X r s , x , θ ∣ 1+|X_r^{s,x,\theta}| 1 + ∣ X r s , x , θ ∣ C s , T C_{s,T} C s , T
1 + ∥ X s , x , θ ∥ ∞ ; [ 0 , T ] ≤ ( 1 + ∣ x ∣ ) exp ( C s , T [ T + TV [ 0 , T ] ( ζ s , θ ) ] ) . 1+\|X^{s,x,\theta}\|_{\infty;[0,T]}\le(1+|x|)\exp\left(C_{s,T}\left[T+\operatorname{TV}_{[0,T]}(\zeta^{s,\theta})\right]\right). 1 + ∥ X s , x , θ ∥ ∞ ; [ 0 , T ] ≤ ( 1 + ∣ x ∣ ) exp ( C s , T [ T + TV [ 0 , T ] ( ζ s , θ ) ] ) . The construction therefore extends to every finite time interval. It also gives measurability without choosing path-dependent time partitions. The set of continuous bounded-variation paths is Borel in C ( [ 0 , T ] ) C([0,T]) C ([ 0 , T ]) x x x θ \theta θ
Young case
For 0 < b ≤ 1 0<b\le1 0 < b ≤ 1 p p p 1 + α > ( H b ∘ ) − 1 1+\alpha>(H_b^\circ)^{-1} 1 + α > ( H b ∘ ) − 1 H b ∘ > 1 / 2 H_b^\circ>1/2 H b ∘ > 1/2 η ∈ ( 1 / p , H b ∘ ) \eta\in(1/p,H_b^\circ) η ∈ ( 1/ p , H b ∘ ) f ( s + ⋅ ) f(s+\cdot) f ( s + ⋅ ) z : = ζ s , 0 z:=\zeta^{s,0} z := ζ s , 0 η \eta η C p -var ( [ 0 , T ] ) C^{p\text{-var}}([0,T]) C p -var ([ 0 , T ]) v : = θ ∣ [ 0 , T ] v:=\theta|_{[0,T]} v := θ ∣ [ 0 , T ] ( y , u ) ↦ ( a ( y , s + u ) , σ ( y , s + u ) ) (y,u)\mapsto(a(y,s+u),\sigma(y,s+u)) ( y , u ) ↦ ( a ( y , s + u ) , σ ( y , s + u )) C p -var ( [ 0 , T ] ) C^{p\text{-var}}([0,T]) C p -var ([ 0 , T ]) ( x , z , v ) (x,z,v) ( x , z , v ) N p N_p N p z z z B T B_T B T v v v ( H b ∘ ) − 1 (H_b^\circ)^{-1} ( H b ∘ ) − 1 p p p p p p
For measurability, let C 0 , p -var ( [ 0 , T ] ) C^{0,p\text{-var}}([0,T]) C 0 , p -var ([ 0 , T ]) N p N_p N p [9 Section 2.1, following (2.1)] . For 1 / p < η 1 < η 1/p<\eta_1<\eta 1/ p < η 1 < η z ( n ) z^{(n)} z ( n ) T 2 − n T2^{-n} T 2 − n
∥ z ( n ) − z ∥ C η 1 ( [ 0 , T ] ) ≤ C η , η 1 , T 2 − n ( η − η 1 ) [ z ] η ; [ 0 , T ] \|z^{(n)}-z\|_{C^{\eta_1}([0,T])}\le C_{\eta,\eta_1,T}2^{-n(\eta-\eta_1)}[z]_{\eta;[0,T]} ∥ z ( n ) − z ∥ C η 1 ([ 0 , T ]) ≤ C η , η 1 , T 2 − n ( η − η 1 ) [ z ] η ; [ 0 , T ] follows from ∥ z ( n ) − z ∥ ∞ ≤ C ( T 2 − n ) η [ z ] η \|z^{(n)}-z\|_\infty\le C(T2^{-n})^\eta[z]_\eta ∥ z ( n ) − z ∥ ∞ ≤ C ( T 2 − n ) η [ z ] η [ z ( n ) − z ] η ≤ C [ z ] η [z^{(n)}-z]_\eta\le C[z]_\eta [ z ( n ) − z ] η ≤ C [ z ] η z ( n ) → z z^{(n)}\to z z ( n ) → z N p N_p N p C 0 , p -var C^{0,p\text{-var}} C 0 , p -var L 1 L^1 L 1 N p N_p N p z z z z z z C 0 , p -var ( [ 0 , T ] ) C^{0,p\text{-var}}([0,T]) C 0 , p -var ([ 0 , T ]) θ ↦ θ ∣ [ 0 , T ] \theta\mapsto\theta|_{[0,T]} θ ↦ θ ∣ [ 0 , T ] B T B_T B T ( x , θ , z ) ∈ R × E γ × C 0 , p -var ( [ 0 , T ] ) (x,\theta,z)\in\mathbb{R}\times E_\gamma\times C^{0,p\text{-var}}([0,T]) ( x , θ , z ) ∈ R × E γ × C 0 , p -var ([ 0 , T ])
In the finite-variation case the Picard construction is nonanticipating. In the Young case, the same solution construction on [ 0 , r ] [0,r] [ 0 , r ] [ 0 , T ] [0,T] [ 0 , T ] X X X
Restart and Markov property
For u ≤ t u\le t u ≤ t F F F R × E γ \mathbb{R}\times E_\gamma R × E γ
P u , t F ( x , θ ) : = E [ F ( X t − u u , x , θ , Θ t − u u , θ ) ] . P_{u,t}F(x,\theta):=\mathbb{E}\left[F\left(X_{t-u}^{u,x,\theta},\Theta_{t-u}^{u,\theta}\right)\right]. P u , t F ( x , θ ) := E [ F ( X t − u u , x , θ , Θ t − u u , θ ) ] . By the measurability proved above and the continuity of θ ↦ S r θ \theta\mapsto S_r\theta θ ↦ S r θ 0 ≤ r ≤ T 0\le r\le T 0 ≤ r ≤ T r r r
Θ r + h s , θ = S h Θ r s , θ + Ξ h s + r , ( r ) , 0 ≤ h ≤ T − r , \Theta_{r+h}^{s,\theta}=S_h\Theta_r^{s,\theta}+\Xi_h^{s+r,(r)},\qquad0\le h\le T-r, Θ r + h s , θ = S h Θ r s , θ + Ξ h s + r , ( r ) , 0 ≤ h ≤ T − r , simultaneously for all h h h Ξ s + r , ( r ) \Xi^{s+r,(r)} Ξ s + r , ( r ) f ( s + r + ⋅ ) f(s+r+\cdot) f ( s + r + ⋅ ) W r + ⋅ − W r W_{r+\cdot}-W_r W r +⋅ − W r F r \mathcal{F}_r F r h h h
ζ ^ h : = Θ r s , θ ( h ) + Ξ h s + r , ( r ) ( 0 ) = ζ r + h s , θ . \widehat{\zeta}_h:=\Theta_r^{s,\theta}(h)+\Xi_h^{s+r,(r)}(0)=\zeta_{r+h}^{s,\theta}. ζ h := Θ r s , θ ( h ) + Ξ h s + r , ( r ) ( 0 ) = ζ r + h s , θ . The first summand is the restriction of the new state to its horizon variable, and has bounded variation by (50) . The second is a fresh Gaussian path with the regularity used above. Additivity and time translation of the pathwise integrals show that X ^ h : = X r + h s , x , θ \widehat{X}_h := X_{r+h}^{s,x,\theta} X h := X r + h s , x , θ
X ^ h = X r s , x , θ + ∫ 0 h a ( X ^ u , s + r + u ) d u + ∫ 0 h σ ( X ^ u , s + r + u ) d ζ ^ u . \widehat{X}_h = X_r^{s,x,\theta} + \int_0^h a(\widehat{X}_u,s+r+u)\,\mathrm{d}u + \int_0^h \sigma(\widehat{X}_u,s+r+u)\,\mathrm{d}\widehat{\zeta}_u. X h = X r s , x , θ + ∫ 0 h a ( X u , s + r + u ) d u + ∫ 0 h σ ( X u , s + r + u ) d ζ u . By uniqueness, this is the restarted solution from X r s , x , θ X_r^{s,x,\theta} X r s , x , θ s + r s+r s + r F r \mathcal{F}_r F r F r \mathcal{F}_r F r
E [ F ( X q s , x , θ ) | F r ] = P s + r , s + q F ( X r s , x , θ ) \mathbb{E}\left[F\left(X_q^{s,x,\theta}\right)\middle|\mathcal{F}_r\right] = P_{s+r,s+q}F\left(X_r^{s,x,\theta}\right) E [ F ( X q s , x , θ ) F r ] = P s + r , s + q F ( X r s , x , θ ) almost surely, $0 \le r \le q \le T$.
Applying this at an intermediate time also gives P u , t = P u , v P v , t P_{u,t}=P_{u,v}P_{v,t} P u , t = P u , v P v , t u ≤ v ≤ t u\le v\le t u ≤ v ≤ t
Continuity in the lifted state
Let ( x n , θ n ) → ( x , θ ) (x_n,\theta_n)\to(x,\theta) ( x n , θ n ) → ( x , θ ) R × E γ \mathbb{R}\times E_\gamma R × E γ Θ r s , θ n − Θ r s , θ = S r ( θ n − θ ) \Theta_r^{s,\theta_n}-\Theta_r^{s,\theta}=S_r(\theta_n-\theta) Θ r s , θ n − Θ r s , θ = S r ( θ n − θ ) (50) the curves converge in E γ E_\gamma E γ [ 0 , T ] [0,T] [ 0 , T ] B T ( θ n − θ ) → 0 B_T(\theta_n-\theta)\to0 B T ( θ n − θ ) → 0
If 1 < b < 2 1<b<2 1 < b < 2 K : = sup n TV [ 0 , T ] ( θ n ) < ∞ K:=\sup_n\operatorname{TV}_{[0,T]}(\theta_n)<\infty K := sup n TV [ 0 , T ] ( θ n ) < ∞ D n ( r ) : = ∣ X r s , x n , θ n − X r s , x , θ ∣ D_n(r):=\left|X_r^{s,x_n,\theta_n}-X_r^{s,x,\theta}\right| D n ( r ) := X r s , x n , θ n − X r s , x , θ ζ s , 0 + θ n \zeta^{s,0}+\theta_n ζ s , 0 + θ n θ n − θ \theta_n-\theta θ n − θ S : = sup r ≤ T ∣ σ ( X r s , x , θ , s + r ) ∣ < ∞ S:=\sup_{r\le T}|\sigma(X_r^{s,x,\theta},s+r)|<\infty S := sup r ≤ T ∣ σ ( X r s , x , θ , s + r ) ∣ < ∞
D n ( r ) ≤ ∣ x n − x ∣ + S TV [ 0 , T ] ( θ n − θ ) + L T ∫ 0 r D n ( u ) ( d u + ∣ d ζ u s , 0 ∣ + ∣ d θ n ( u ) ∣ ) . D_n(r)\le|x_n-x|+S\operatorname{TV}_{[0,T]}(\theta_n-\theta)+L_T\int_0^r D_n(u)\left(\mathrm{d}u+|\mathrm{d}\zeta_u^{s,0}|+|\mathrm{d}\theta_n(u)|\right). D n ( r ) ≤ ∣ x n − x ∣ + S TV [ 0 , T ] ( θ n − θ ) + L T ∫ 0 r D n ( u ) ( d u + ∣ d ζ u s , 0 ∣ + ∣ d θ n ( u ) ∣ ) . The atomless-measure Gronwall inequality therefore yields
sup r ≤ T ∣ X r s , x n , θ n − X r s , x , θ ∣ ≤ exp ( L T [ T + TV [ 0 , T ] ( ζ s , 0 ) + K ] ) ( ∣ x n − x ∣ + S TV [ 0 , T ] ( θ n − θ ) ) ⟶ 0. \sup_{r\le T}\left|X_r^{s,x_n,\theta_n}-X_r^{s,x,\theta}\right|
\le\exp\left(L_T\left[T+\operatorname{TV}_{[0,T]}(\zeta^{s,0})+K\right]\right)
\left(|x_n-x|+S\operatorname{TV}_{[0,T]}(\theta_n-\theta)\right)\longrightarrow0. r ≤ T sup X r s , x n , θ n − X r s , x , θ ≤ exp ( L T [ T + TV [ 0 , T ] ( ζ s , 0 ) + K ] ) ( ∣ x n − x ∣ + S TV [ 0 , T ] ( θ n − θ ) ) ⟶ 0. If 0 < b ≤ 1 0<b\le1 0 < b ≤ 1 z = ζ s , 0 z=\zeta^{s,0} z = ζ s , 0 B T ( θ n − θ ) → 0 B_T(\theta_n-\theta)\to0 B T ( θ n − θ ) → 0 F ∈ C b ( R × E γ ) F\in C_b(\mathbb{R}\times E_\gamma) F ∈ C b ( R × E γ ) P u , t F P_{u,t}F P u , t F ∥ P u , t F ∥ ∞ ≤ ∥ F ∥ ∞ \|P_{u,t}F\|_\infty\le\|F\|_\infty ∥ P u , t F ∥ ∞ ≤ ∥ F ∥ ∞ C b ( R × E γ ) C_b(\mathbb{R}\times E_\gamma) C b ( R × E γ )
Finally, for s = 0 s=0 s = 0 θ = 0 \theta=0 θ = 0 (49) reduces to (30) , so ζ 0 , 0 \zeta^{0,0} ζ 0 , 0 (31) . □
Remark 3.11 For κ > 0 \kappa>0 κ > 0 σ 0 ∈ R \sigma_0\in\mathbb{R} σ 0 ∈ R a ( x , t ) = − κ x a(x,t)=-\kappa x a ( x , t ) = − κ x σ ( x , t ) = σ 0 \sigma(x,t)=\sigma_0 σ ( x , t ) = σ 0
X r s , x , θ = e − κ r x + σ 0 ∫ 0 r e − κ ( r − u ) d ζ u s , θ . X_r^{s,x,\theta}=e^{-\kappa r}x+\sigma_0\int_0^r e^{-\kappa(r-u)}\,\mathrm{d}\zeta_u^{s,\theta}. X r s , x , θ = e − κ r x + σ 0 ∫ 0 r e − κ ( r − u ) d ζ u s , θ . where the integral has the Young or Lebesgue–Stieltjes interpretation specified in Theorem 3.10 . For linear multiplicative diffusion, a ( x , t ) = ν x a(x,t)=\nu x a ( x , t ) = ν x σ ( x , t ) = β x \sigma(x,t)=\beta x σ ( x , t ) = β x ν , β ∈ R \nu,\beta\in\mathbb{R} ν , β ∈ R
X r s , x , θ = x exp ( ν r + β [ ζ r s , θ − ζ 0 s , θ ] ) . X_r^{s,x,\theta}=x\exp\left(\nu r+\beta\left[\zeta_r^{s,\theta}-\zeta_0^{s,\theta}\right]\right). X r s , x , θ = x exp ( ν r + β [ ζ r s , θ − ζ 0 s , θ ] ) . Both formulas follow from the pathwise integration-by-parts and chain rules. The driving paths are continuous and have finite p p p p < 2 p<2 p < 2
Spectral Ornstein–Uhlenbeck representation and multifactor approximation
The Laplace factorization expresses the boundary process as a superposition of integrated Ornstein–Uhlenbeck factors. Retaining finitely many relaxation rates gives a finite-dimensional Markovian approximation. We couple it with the original process, estimate the spectral error in the path norms needed by the nonlinear equation, and then control the additional error from time discretization.
Throughout this section, f f f [ 0 , T ] [0,T] [ 0 , T ] f f f
Spaces and notation.
We use U b = L 2 ( ( 0 , ∞ ) , μ b ) \mathcal{U}_b=L^2((0,\infty),\mu_b) U b = L 2 (( 0 , ∞ ) , μ b ) λ \lambda λ E γ \mathcal{E}_\gamma E γ r r r t ↦ ζ t , f , b t\mapsto\zeta_{t,f,b} t ↦ ζ t , f , b 0 < γ < 1 0<\gamma<1 0 < γ < 1 b + γ > 1 b+\gamma>1 b + γ > 1 R × R M \mathbb{R}\times\mathbb{R}^M R × R M
For a separable Banach space E E E 1 ≤ p < ∞ 1\leq p<\infty 1 ≤ p < ∞ L p ( Ω ; E ) L^p(\Omega;E) L p ( Ω ; E ) E E E
∥ U ∥ L p ( Ω ; E ) : = ( E ∥ U ∥ E p ) 1 / p . \lVert U\rVert_{L^p(\Omega;E)}:=\left(\mathbb{E}\lVert U\rVert_E^p\right)^{1/p}. ∥ U ∥ L p ( Ω ; E ) := ( E ∥ U ∥ E p ) 1/ p . The Bochner space L q ( 0 , T ; E ) L^q(0,T;E) L q ( 0 , T ; E ) 1 ≤ q < ∞ 1\leq q<\infty 1 ≤ q < ∞ ( ∫ 0 T ∥ u ( t ) ∥ E q d t ) 1 / q \left(\int_0^T\lVert u(t)\rVert_E^q\,\mathrm{d}t\right)^{1/q} ( ∫ 0 T ∥ u ( t ) ∥ E q d t ) 1/ q E E E E = R E=\mathbb{R} E = R
∥ D ∥ L p ( Ω ; L q ( 0 , T ) ) : = ( E [ ( ∫ 0 T ∣ D t ∣ q d t ) p / q ] ) 1 / p . \lVert D\rVert_{L^p(\Omega;L^q(0,T))}:=\left(\mathbb{E}\left[\left(\int_0^T\lvert D_t\rvert^q\,\mathrm{d}t\right)^{p/q}\right]\right)^{1/p}. ∥ D ∥ L p ( Ω ; L q ( 0 , T )) := E ( ∫ 0 T ∣ D t ∣ q d t ) p / q 1/ p . For 1 < b < 2 1<b<2 1 < b < 2 L 2 ( 0 , T ) L^2(0,T) L 2 ( 0 , T ) 0 < b ≤ 1 0<b\leq1 0 < b ≤ 1 C γ ( [ 0 , T ] ) C^\gamma([0,T]) C γ ([ 0 , T ]) c γ ( [ 0 , T ] ) c^\gamma([0,T]) c γ ([ 0 , T ]) C p -var ( [ 0 , T ] ) C^{p\text{-var}}([0,T]) C p -var ([ 0 , T ]) N p N_p N p
From Laplace features to Ornstein–Uhlenbeck factors
For λ > 0 \lambda>0 λ > 0
k λ ( r ) : = 1 − e − λ r λ = ∫ 0 r e − λ v d v . k_\lambda(r):=\frac{1-e^{-\lambda r}}{\lambda}=\int_0^r e^{-\lambda v}\,\mathrm{d}v. k λ ( r ) := λ 1 − e − λ r = ∫ 0 r e − λ v d v . For every 0 < b < 2 0<b<2 0 < b < 2 b = 1 b=1 b = 1 ϕ r ( λ ) = b k λ ( r ) \phi_r(\lambda)=\sqrt{b}\,k_\lambda(r) ϕ r ( λ ) = b k λ ( r ) W W W W W W μ b ( d λ ) d u \mu_b(\mathrm{d}\lambda)\,\mathrm{d}u μ b ( d λ ) d u
W t ( h ) = ∫ 0 t ∫ 0 ∞ h ( λ ) W ( d λ , d u ) , h ∈ U b . W_t(h)=\int_0^t\int_0^\infty h(\lambda)W(\mathrm{d}\lambda,\mathrm{d}u),\qquad h\in U_b. W t ( h ) = ∫ 0 t ∫ 0 ∞ h ( λ ) W ( d λ , d u ) , h ∈ U b . The boundary process has the representation
ζ t , f , b = b ∫ 0 t ∫ 0 ∞ 1 − e − λ ( t − u ) λ f ( u ) W ( d λ , d u ) . (51) \zeta_{t,f,b}=\sqrt{b}\int_0^t\int_0^\infty\frac{1-e^{-\lambda(t-u)}}{\lambda}\sqrt{f(u)}W(\mathrm{d}\lambda,\mathrm{d}u). \tag*{(51)} ζ t , f , b = b ∫ 0 t ∫ 0 ∞ λ 1 − e − λ ( t − u ) f ( u ) W ( d λ , d u ) . (51) The integrated kernel is square integrable over the full spectrum. Indeed, 0 ≤ k λ ( r ) ≤ min { r , λ − 1 } 0\leq k_\lambda(r)\leq\min\{r,\lambda^{-1}\} 0 ≤ k λ ( r ) ≤ min { r , λ − 1 } μ b ( d λ ) = λ 1 − b d λ / Γ ( 2 − b ) \mu_b(\mathrm{d}\lambda)=\lambda^{1-b}\mathrm{d}\lambda/\Gamma(2-b) μ b ( d λ ) = λ 1 − b d λ /Γ ( 2 − b ) λ = r − 1 \lambda=r^{-1} λ = r − 1
∫ 0 ∞ k λ ( r ) 2 μ b ( d λ ) ≤ 1 Γ ( 2 − b ) ( r 2 ∫ 0 1 / r λ 1 − b d λ + ∫ 1 / r ∞ λ − 1 − b d λ ) ≤ C b r b , r > 0. \int_0^\infty k_\lambda(r)^2\mu_b(\mathrm{d}\lambda)\leq\frac{1}{\Gamma(2-b)}\left(r^2\int_0^{1/r}\lambda^{1-b}\mathrm{d}\lambda+\int_{1/r}^\infty\lambda^{-1-b}\mathrm{d}\lambda\right)\leq C_b r^b,\qquad r>0. ∫ 0 ∞ k λ ( r ) 2 μ b ( d λ ) ≤ Γ ( 2 − b ) 1 ( r 2 ∫ 0 1/ r λ 1 − b d λ + ∫ 1/ r ∞ λ − 1 − b d λ ) ≤ C b r b , r > 0. Consequently the variance in (51) is bounded by C b ∫ 0 t f ( u ) ( t − u ) b d u C_b\int_0^t f(u)(t-u)^b\,\mathrm{d}u C b ∫ 0 t f ( u ) ( t − u ) b d u 0 < b < 2 0<b<2 0 < b < 2 b = 1 b=1 b = 1
For h ∈ U b h\in U_b h ∈ U b
M t ( h ) : = ∫ 0 t ∫ 0 ∞ h ( λ ) f ( u ) W ( d λ , d u ) , M_t(h):=\int_0^t\int_0^\infty h(\lambda)\sqrt{f(u)}W(\mathrm{d}\lambda,\mathrm{d}u), M t ( h ) := ∫ 0 t ∫ 0 ∞ h ( λ ) f ( u ) W ( d λ , d u ) , and
Y t ( h ) : = ∫ 0 t ∫ 0 ∞ h ( λ ) e − λ ( t − u ) f ( u ) W ( d λ , d u ) . Y_t(h):=\int_0^t\int_0^\infty h(\lambda)e^{-\lambda(t-u)}\sqrt{f(u)}W(\mathrm{d}\lambda,\mathrm{d}u). Y t ( h ) := ∫ 0 t ∫ 0 ∞ h ( λ ) e − λ ( t − u ) f ( u ) W ( d λ , d u ) . For each t t t U b U_b U b L 2 ( Ω ) L^2(\Omega) L 2 ( Ω )
More precisely, the Itô isometry gives
max { ∥ M t ( h ) ∥ L 2 ( Ω ) , ∥ Y t ( h ) ∥ L 2 ( Ω ) } ≤ ∥ h ∥ U b ( ∫ 0 t f ( u ) d u ) 1 / 2 . \max\left\{\lVert M_t(h)\rVert_{L^2(\Omega)},\lVert Y_t(h)\rVert_{L^2(\Omega)}\right\}\leq\lVert h\rVert_{U_b}\left(\int_0^t f(u)\,\mathrm{d}u\right)^{1/2}. max { ∥ M t ( h ) ∥ L 2 ( Ω ) , ∥ Y t ( h ) ∥ L 2 ( Ω ) } ≤ ∥ h ∥ U b ( ∫ 0 t f ( u ) d u ) 1/2 . These scalar evaluations specify the cylindrical processes. The spectral space U b U_b U b E γ E_\gamma E γ U b U_b U b U b U_b U b
( D b h ) ( λ ) : = λ h ( λ ) , D ( D b ) : = { h ∈ U b : λ h ( λ ) ∈ U b } . (D_bh)(\lambda):=\lambda h(\lambda),\qquad D(D_b):=\{h\in U_b:\lambda h(\lambda)\in U_b\}. ( D b h ) ( λ ) := λh ( λ ) , D ( D b ) := { h ∈ U b : λh ( λ ) ∈ U b } . The operator D b D_b D b λ \lambda λ A A A E γ E_\gamma E γ h ∈ D ( D b ) h\in D(D_b) h ∈ D ( D b )
∫ 0 t ( ∫ 0 s f ( u ) ∥ e − ⋅ ( s − u ) D b h ∥ U b 2 d u ) 1 / 2 d s ≤ t ∥ D b h ∥ U b ( ∫ 0 t f ( u ) d u ) 1 / 2 < ∞ . \int_{0}^{t}\left(\int_{0}^{s}f(u)\left\|e^{-\cdot(s-u)}D_bh\right\|_{U_b}^{2}\,du\right)^{1/2}ds\le t\left\|D_bh\right\|_{U_b}\left(\int_{0}^{t}f(u)\,du\right)^{1/2}<\infty. ∫ 0 t ( ∫ 0 s f ( u ) e −⋅ ( s − u ) D b h U b 2 d u ) 1/2 d s ≤ t ∥ D b h ∥ U b ( ∫ 0 t f ( u ) d u ) 1/2 < ∞. Stochastic Fubini [11 Section 4.5] therefore yields the weak Ornstein–Uhlenbeck identity
Y t ( h ) = M t ( h ) − ∫ 0 t Y s ( D b h ) d s . Y_t(h)=M_t(h)-\int_{0}^{t}Y_s(D_bh)\,ds. Y t ( h ) = M t ( h ) − ∫ 0 t Y s ( D b h ) d s . This identity defines the cylindrical Ornstein–Uhlenbeck dynamics through their scalar evaluations on U b U_b U b
Since μ b ( ( 0 , ∞ ) ) = ∞ \mu_b((0,\infty))=\infty μ b (( 0 , ∞ )) = ∞ 1 1 1 U b U_b U b h = 1 h=1 h = 1
If 1 < b < 2 1<b<2 1 < b < 2 f ∈ L l o c ∞ ( R + ) f\in L^\infty_{\mathrm{loc}}(\mathbb{R}_+) f ∈ L loc ∞ ( R + ) Proposition 3.4 is
V t = b ∫ 0 t ∫ 0 ∞ e − λ ( t − u ) f ( u ) W ( d λ , d u ) , V_t=\sqrt{b}\int_{0}^{t}\int_{0}^{\infty}e^{-\lambda(t-u)}\sqrt{f(u)}\,W(d\lambda,du), V t = b ∫ 0 t ∫ 0 ∞ e − λ ( t − u ) f ( u ) W ( d λ , d u ) , with variance
E ∣ V t ∣ 2 = b 2 b − 2 ∫ 0 t f ( u ) ( t − u ) b − 2 d u . (52) \mathbb{E}|V_t|^2=b2^{b-2}\int_{0}^{t}f(u)(t-u)^{b-2}\,du. \tag*{(52)} E ∣ V t ∣ 2 = b 2 b − 2 ∫ 0 t f ( u ) ( t − u ) b − 2 d u . (52) Let 0 < b ≤ 1 0<b\le1 0 < b ≤ 1 t > 0 t>0 t > 0 ε ∈ ( 0 , t ] \varepsilon\in(0,t] ε ∈ ( 0 , t ] c > 0 c>0 c > 0 f ( u ) ≥ c f(u)\ge c f ( u ) ≥ c u ∈ ( t − ε , t ) u\in(t-\varepsilon,t) u ∈ ( t − ε , t ) (52) is infinite. The integrated representation (51) is well defined for every 0 < b < 2 0<b<2 0 < b < 2
Fix M ∈ N M\in\mathbb{N} M ∈ N Λ > 0 \Lambda>0 Λ > 0 0 = λ 0 < λ 1 < ⋯ < λ M = Λ 0=\lambda_0<\lambda_1<\cdots<\lambda_M=\Lambda 0 = λ 0 < λ 1 < ⋯ < λ M = Λ k = 1 , … , M k=1,\ldots,M k = 1 , … , M
I k : = ( λ k − 1 , λ k ] , ξ k ∈ I k , I_k:=(\lambda_{k-1},\lambda_k],\qquad\xi_k\in I_k, I k := ( λ k − 1 , λ k ] , ξ k ∈ I k , where I k I_k I k k k k ξ k \xi_k ξ k
δ λ : = max 1 ≤ k ≤ M ( λ k − λ k − 1 ) . \delta_\lambda:=\max_{1\le k\le M}(\lambda_k-\lambda_{k-1}). δ λ := 1 ≤ k ≤ M max ( λ k − λ k − 1 ) . The partitions need not be uniform. For the spectral rates in M M M δ λ ≤ C m e s h Λ / M \delta_\lambda\le C_{\mathrm{mesh}}\Lambda/M δ λ ≤ C mesh Λ/ M C m e s h ≥ 1 C_{\mathrm{mesh}}\ge1 C mesh ≥ 1 M M M Λ \Lambda Λ C m e s h C_{\mathrm{mesh}} C mesh
No lower bound on individual cell widths is imposed, so local refinement is allowed. Define
w k : = μ b ( I k ) = λ k 2 − b − λ k − 1 2 − b ( 2 − b ) Γ ( 2 − b ) , B t k : = w k − 1 / 2 W ( I k × [ 0 , t ] ) . (53) w_k:=\mu_b(I_k)=\frac{\lambda_k^{2-b}-\lambda_{k-1}^{2-b}}{(2-b)\Gamma(2-b)},\qquad B_t^k:=w_k^{-1/2}W(I_k\times[0,t]). \tag*{(53)} w k := μ b ( I k ) = ( 2 − b ) Γ ( 2 − b ) λ k 2 − b − λ k − 1 2 − b , B t k := w k − 1/2 W ( I k × [ 0 , t ]) . (53) The normalized indicators w k − 1 / 2 1 I k w_k^{-1/2}\mathbf{1}_{I_k} w k − 1/2 1 I k U b U_b U b B 1 , … , B M B^1,\ldots,B^M B 1 , … , B M
The weights are exact spectral masses; only the relaxation rates are approximated. Using the same W W W
ϕ r M , Λ ( λ ) : = b ∑ k = 1 M k ξ k ( r ) 1 I k ( λ ) , (54) \phi_r^{M,\Lambda}(\lambda) := \sqrt{b}\sum_{k=1}^{M} k_{\xi_k}(r)\mathbf{1}_{I_k}(\lambda), \tag*{(54)} ϕ r M , Λ ( λ ) := b k = 1 ∑ M k ξ k ( r ) 1 I k ( λ ) , (54) with value zero above Λ \Lambda Λ k k k
Y t k , M , Λ = − ξ k ∫ 0 t Y s k , M , Λ d s + ∫ 0 t f ( s ) d B s k , Y 0 k , M , Λ = 0 , Y_t^{k,M,\Lambda}=-\xi_k\int_0^t Y_s^{k,M,\Lambda}\,\mathrm{d}s+\int_0^t\sqrt{f(s)}\,\mathrm{d}B_s^k,\qquad Y_0^{k,M,\Lambda}=0, Y t k , M , Λ = − ξ k ∫ 0 t Y s k , M , Λ d s + ∫ 0 t f ( s ) d B s k , Y 0 k , M , Λ = 0 , namely
Y t k , M , Λ = ∫ 0 t e − ξ k ( t − s ) f ( s ) d B s k . Y_t^{k,M,\Lambda}=\int_0^t e^{-\xi_k(t-s)}\sqrt{f(s)}\,\mathrm{d}B_s^k. Y t k , M , Λ = ∫ 0 t e − ξ k ( t − s ) f ( s ) d B s k . Define the aggregate factor process
Y t M , Λ : = b ∑ k = 1 M w k Y t k , M , Λ . \mathcal{Y}_t^{M,\Lambda}:=\sqrt{b}\sum_{k=1}^{M}\sqrt{w_k}Y_t^{k,M,\Lambda}. Y t M , Λ := b k = 1 ∑ M w k Y t k , M , Λ . and the finite-factor driver
ζ t M , Λ : = ∫ 0 t Y s M , Λ d s . \zeta_t^{M,\Lambda}:=\int_0^t\mathcal{Y}_s^{M,\Lambda}\,\mathrm{d}s. ζ t M , Λ := ∫ 0 t Y s M , Λ d s . Since f ∈ L 1 ( 0 , t ) f\in L^1(0,t) f ∈ L 1 ( 0 , t )
ζ t M , Λ = b ∑ k = 1 M w k ∫ 0 t k ξ k ( t − u ) f ( u ) d B u k . (55) \zeta_t^{M,\Lambda}=\sqrt{b}\sum_{k=1}^{M}\sqrt{w_k}\int_0^t k_{\xi_k}(t-u)\sqrt{f(u)}\,\mathrm{d}B_u^k. \tag*{(55)} ζ t M , Λ = b k = 1 ∑ M w k ∫ 0 t k ξ k ( t − u ) f ( u ) d B u k . (55) Thus (55) is obtained from (51) by replacing ϕ \phi ϕ ϕ M , Λ \phi^{M,\Lambda} ϕ M , Λ
The corresponding forward curve is
Θ t M , Λ ( r ) : = b ∑ k = 1 M w k ∫ 0 t k ξ k ( t + r − u ) f ( u ) d B u k . \Theta_t^{M,\Lambda}(r):=\sqrt{b}\sum_{k=1}^{M}\sqrt{w_k}\int_0^t k_{\xi_k}(t+r-u)\sqrt{f(u)}\,\mathrm{d}B_u^k. Θ t M , Λ ( r ) := b k = 1 ∑ M w k ∫ 0 t k ξ k ( t + r − u ) f ( u ) d B u k . The identity k λ ( a + r ) = k λ ( a ) + e − λ a k λ ( r ) k_\lambda(a+r)=k_\lambda(a)+e^{-\lambda a}k_\lambda(r) k λ ( a + r ) = k λ ( a ) + e − λa k λ ( r )
Θ t M , Λ ( r ) = ζ t M , Λ + b ∑ k = 1 M w k k ξ k ( r ) Y t k , M , Λ . (56) \Theta_t^{M,\Lambda}(r)=\zeta_t^{M,\Lambda}+\sqrt{b}\sum_{k=1}^{M}\sqrt{w_k}k_{\xi_k}(r)Y_t^{k,M,\Lambda}. \tag*{(56)} Θ t M , Λ ( r ) = ζ t M , Λ + b k = 1 ∑ M w k k ξ k ( r ) Y t k , M , Λ . (56) Its derivative is a finite sum of decaying exponentials, so Θ t M , Λ ∈ E γ \Theta_t^{M,\Lambda} \in\mathcal{E}_\gamma Θ t M , Λ ∈ E γ t t t r ≥ 0 r \ge0 r ≥ 0
ζ t + r M , Λ − ζ t M , Λ = b ∑ k = 1 M ω k ( h ξ k ( r ) Y t k , M , Λ + ∫ t t + r h ξ k ( t + r − u ) f ( u ) d B u k ) . \zeta_{t+r}^{M,\Lambda}-\zeta_t^{M,\Lambda}=\sqrt{b}\sum_{k=1}^{M}\sqrt{\omega_k}\left(h_{\xi_k}(r)Y_t^{k,M,\Lambda}+\int_t^{t+r}h_{\xi_k}(t+r-u)\sqrt{f(u)}\,\mathrm{d}B_u^k\right). ζ t + r M , Λ − ζ t M , Λ = b k = 1 ∑ M ω k ( h ξ k ( r ) Y t k , M , Λ + ∫ t t + r h ξ k ( t + r − u ) f ( u ) d B u k ) . The first term is determined by the current factors and equals Θ t M , Λ ( r ) − Θ t M , Λ ( 0 ) \Theta_t^{M,\Lambda}(r)-\Theta_t^{M,\Lambda}(0) Θ t M , Λ ( r ) − Θ t M , Λ ( 0 ) M M M
For t > 0 t>0 t > 0 g ∈ C ( [ 0 , t ] ) g\in C([0,t]) g ∈ C ([ 0 , t ])
∫ 0 t g ( s ) d ζ s M , Λ = ∫ 0 t g ( s ) Y s M , Λ d s . \int_0^t g(s)\,\mathrm{d}\zeta_s^{M,\Lambda}=\int_0^t g(s)Y_s^{M,\Lambda}\,\mathrm{d}s. ∫ 0 t g ( s ) d ζ s M , Λ = ∫ 0 t g ( s ) Y s M , Λ d s . If the coefficients are continuous and globally Lipschitz in x x x
X t M , Λ = x 0 + ∫ 0 t a ( X s M , Λ , s ) d s + ∫ 0 t σ ( X s M , Λ , s ) Y s M , Λ d s X_t^{M,\Lambda}=x_0+\int_0^t a(X_s^{M,\Lambda},s)\,\mathrm{d}s+\int_0^t \sigma(X_s^{M,\Lambda},s)Y_s^{M,\Lambda}\,\mathrm{d}s X t M , Λ = x 0 + ∫ 0 t a ( X s M , Λ , s ) d s + ∫ 0 t σ ( X s M , Λ , s ) Y s M , Λ d s has a unique global solution. These conditions hold under Assumption 3.5. Set
Y t M , Λ : = ( Y t 1 , M , Λ , … , Y t M , M , Λ ) ∈ R M . \mathbf{Y}_t^{M,\Lambda}:=(Y_t^{1,M,\Lambda},\ldots,Y_t^{M,M,\Lambda})\in\mathbb{R}^M. Y t M , Λ := ( Y t 1 , M , Λ , … , Y t M , M , Λ ) ∈ R M . Then ( X M , Λ , Y M , Λ ) (X^{M,\Lambda},\mathbf{Y}^{M,\Lambda}) ( X M , Λ , Y M , Λ ) R × R M \mathbb{R}\times\mathbb{R}^M R × R M C b C_b C b Proposition 4.8 establishes this property for arbitrary initial states.
The finite-variation rate
For 1 < b < 2 1<b<2 1 < b < 2 V − Y M , Λ V-Y^{M,\Lambda} V − Y M , Λ
TV [ 0 , T ] ( ζ f , b − ζ M , Λ ) = ∫ 0 T ∣ V t − Y t M , Λ ∣ d t ≤ T ∥ V − Y M , Λ ∥ L 2 ( 0 , T ) . \operatorname{TV}_{[0,T]}(\zeta_{f,b}-\zeta^{M,\Lambda})=\int_0^T\left|V_t-Y_t^{M,\Lambda}\right|\,\mathrm{d}t\le\sqrt{T}\left\|V-Y^{M,\Lambda}\right\|_{L^2(0,T)}. TV [ 0 , T ] ( ζ f , b − ζ M , Λ ) = ∫ 0 T V t − Y t M , Λ d t ≤ T V − Y M , Λ L 2 ( 0 , T ) . An L 2 ( 0 , T ) L^2(0,T) L 2 ( 0 , T )
Theorem 4.1 Let 1 < b < 2 1 < b < 2 1 < b < 2 α : = 2 − b ∈ ( 0 , 1 ) \alpha:= 2-b \in(0,1) α := 2 − b ∈ ( 0 , 1 ) T > 0 T > 0 T > 0 M ∈ N M \in\mathbb{N} M ∈ N Λ > 0 \Lambda> 0 Λ > 0 f ∈ L ∞ ( 0 , T ) f \in L^\infty(0,T) f ∈ L ∞ ( 0 , T ) [ 0 , Λ ] [0,\Lambda] [ 0 , Λ ] δ λ \delta_\lambda δ λ
E ∫ 0 T ∣ V t − Y t M , Λ ∣ 2 d t ≤ b ∥ f ∥ ∞ , T 2 T 4 12 α Γ ( α ) δ λ 2 Λ α + b ∥ f ∥ ∞ , T 2 T 2 ( 1 − α ) Γ ( α ) Λ α − 1 . (57) \mathbb{E}\int_0^T \left|V_t-\mathcal{Y}^{M,\Lambda}_t\right|^2\,\mathrm{d}t
\le\frac{b\|f\|_{\infty,T}^2T^4}{12\alpha\Gamma(\alpha)}\delta_\lambda^2\Lambda^\alpha
+\frac{b\|f\|_{\infty,T}^2T}{2(1-\alpha)\Gamma(\alpha)}\Lambda^{\alpha-1}. \tag*{(57)} E ∫ 0 T V t − Y t M , Λ 2 d t ≤ 12 α Γ ( α ) b ∥ f ∥ ∞ , T 2 T 4 δ λ 2 Λ α + 2 ( 1 − α ) Γ ( α ) b ∥ f ∥ ∞ , T 2 T Λ α − 1 . (57) Set Λ = M 2 / 3 \Lambda=M^{2/3} Λ = M 2/3 δ λ ≤ C m e s h M − 1 / 3 \delta_\lambda\le C_{\mathrm{mesh}}M^{-1/3} δ λ ≤ C mesh M − 1/3 p ≥ 2 p\ge2 p ≥ 2
( E ∥ V − Y M , Λ ∥ L 2 ( 0 , T ) p ) 1 / p ≤ C M − ( b − 1 ) / 3 . (58) \left(\mathbb{E}\left\|V-\mathcal{Y}^{M,\Lambda}\right\|_{L^2(0,T)}^p\right)^{1/p}
\le CM^{-(b-1)/3}. \tag*{(58)} ( E V − Y M , Λ L 2 ( 0 , T ) p ) 1/ p ≤ C M − ( b − 1 ) /3 . (58) and
( E sup t ≤ T ∣ ζ t , f , b − ζ t , f M , Λ ∣ p ) 1 / p ≤ C M − ( b − 1 ) / 3 . (59) \left(\mathbb{E}\sup_{t\le T}\left|\zeta_{t,f,b}-\zeta^{M,\Lambda}_{t,f}\right|^p\right)^{1/p}
\le CM^{-(b-1)/3}. \tag*{(59)} ( E t ≤ T sup ζ t , f , b − ζ t , f M , Λ p ) 1/ p ≤ C M − ( b − 1 ) /3 . (59) If, in addition, Assumption 3.5 holds, the coupled solutions with the same deterministic initial value x 0 x_0 x 0
( E sup t ≤ T ∣ X t − X t M , Λ ∣ p ) 1 / p ≤ C M − ( b − 1 ) / 3 . (60) \left(\mathbb{E}\sup_{t\le T}\left|X_t-X^{M,\Lambda}_t\right|^p\right)^{1/p}
\le CM^{-(b-1)/3}. \tag*{(60)} ( E t ≤ T sup X t − X t M , Λ p ) 1/ p ≤ C M − ( b − 1 ) /3 . (60) Proof. Under the coupling induced by W W W
E ∣ V t − Y t M , Λ ∣ 2 = b ∫ 0 t f ( u ) ∑ k = 1 M ∫ I k ∣ e − λ ( t − u ) − e − ξ k ( t − u ) ∣ 2 μ b ( d λ ) d u + b ∫ 0 t f ( u ) ∫ Λ ∞ e − 2 λ ( t − u ) μ b ( d λ ) d u . \mathbb{E}\left|V_t-\mathcal{Y}^{M,\Lambda}_t\right|^2
=b\int_0^t f(u)\sum_{k=1}^M\int_{I_k}\left|e^{-\lambda(t-u)}-e^{-\xi_k(t-u)}\right|^2\mu_b(\mathrm{d}\lambda)\,\mathrm{d}u
+b\int_0^t f(u)\int_\Lambda^\infty e^{-2\lambda(t-u)}\mu_b(\mathrm{d}\lambda)\,\mathrm{d}u. E V t − Y t M , Λ 2 = b ∫ 0 t f ( u ) k = 1 ∑ M ∫ I k e − λ ( t − u ) − e − ξ k ( t − u ) 2 μ b ( d λ ) d u + b ∫ 0 t f ( u ) ∫ Λ ∞ e − 2 λ ( t − u ) μ b ( d λ ) d u . For the quadrature error, use ∣ e − λ r − e − ξ r ∣ ≤ r δ λ \left|e^{-\lambda r}-e^{-\xi r}\right|\le r\delta_\lambda e − λ r − e − ξ r ≤ r δ λ
μ b ( [ 0 , Λ ] ) = Λ α α Γ ( α ) . \mu_b([0,\Lambda])=\frac{\Lambda^\alpha}{\alpha\Gamma(\alpha)}. μ b ([ 0 , Λ ]) = α Γ ( α ) Λ α . Integrating over 0 ≤ u ≤ t ≤ T 0\le u\le t\le T 0 ≤ u ≤ t ≤ T ∫ 0 T ∫ 0 t ( t − u ) 2 d u d t = T 4 / 12 \int_0^T\int_0^t(t-u)^2\,\mathrm{d}u\,\mathrm{d}t=T^4/12 ∫ 0 T ∫ 0 t ( t − u ) 2 d u d t = T 4 /12
For the tail, integrate first in t t t ∫ 0 T − u e − 2 λ r d r ≤ ( 2 λ ) − 1 \int_0^{T-u}e^{-2\lambda r}\,\mathrm{d}r\le(2\lambda)^{-1} ∫ 0 T − u e − 2 λ r d r ≤ ( 2 λ ) − 1
b ∥ f ∥ ∞ , T T 2 Γ ( α ) ∫ Λ ∞ λ α − 2 d λ = b ∥ f ∥ ∞ , T T 2 ( 1 − α ) Γ ( α ) Λ α − 1 , \frac{b\|f\|_{\infty,T}T}{2\Gamma(\alpha)}\int_\Lambda^\infty\lambda^{\alpha-2}\,\mathrm{d}\lambda
=\frac{b\|f\|_{\infty,T}T}{2(1-\alpha)\Gamma(\alpha)}\Lambda^{\alpha-1}, 2Γ ( α ) b ∥ f ∥ ∞ , T T ∫ Λ ∞ λ α − 2 d λ = 2 ( 1 − α ) Γ ( α ) b ∥ f ∥ ∞ , T T Λ α − 1 , where α < 1 \alpha<1 α < 1 δ λ ≤ C m e s h Λ / M \delta_\lambda\le C_{\mathrm{mesh}}\Lambda/M δ λ ≤ C mesh Λ/ M C M − 2 Λ α + 2 CM^{-2}\Lambda^{\alpha+2} C M − 2 Λ α + 2 Λ 3 = M 2 \Lambda^3=M^2 Λ 3 = M 2 Λ = M 2 / 3 \Lambda=M^{2/3} Λ = M 2/3 M − 2 ( b − 1 ) / 3 M^{-2(b-1)/3} M − 2 ( b − 1 ) /3
Λ M M − 1 ⟶ ∞ , δ λ , M 2 Λ M 2 − b ⟶ 0. \Lambda_M M^{-1}\longrightarrow\infty,\qquad\delta_{\lambda,M}^2\Lambda_M^{2-b}\longrightarrow0. Λ M M − 1 ⟶ ∞ , δ λ , M 2 Λ M 2 − b ⟶ 0. The powers of M M M G G G E ∫ 0 T ∣ G t ∣ 2 d t < ∞ \mathbb{E}\int_0^T|G_t|^2\,\mathrm{d}t<\infty E ∫ 0 T ∣ G t ∣ 2 d t < ∞ p ≥ 2 p\ge2 p ≥ 2
( E ∥ G ∥ L 2 ( 0 , T ) p ) 2 / p ≤ ∫ 0 T ( E ∣ G t ∣ p ) 2 / p d t = m p 2 E ∫ 0 T ∣ G t ∣ 2 d t , m p : = ( E ∣ N ∣ p ) 1 / p , \left(\mathbb{E}\|G\|_{L^2(0,T)}^p\right)^{2/p}
\le\int_0^T\left(\mathbb{E}|G_t|^p\right)^{2/p}\,\mathrm{d}t
=m_p^2\mathbb{E}\int_0^T|G_t|^2\,\mathrm{d}t,\qquad m_p:=\left(\mathbb{E}|N|^p\right)^{1/p}, ( E ∥ G ∥ L 2 ( 0 , T ) p ) 2/ p ≤ ∫ 0 T ( E ∣ G t ∣ p ) 2/ p d t = m p 2 E ∫ 0 T ∣ G t ∣ 2 d t , m p := ( E ∣ N ∣ p ) 1/ p , where N N N G = V − Y M , Λ G=V-\mathcal{Y}^{M,\Lambda} G = V − Y M , Λ (58) .
The total-variation estimate at the start of this subsection also controls the uniform norm, since the driver error vanishes at zero. It therefore gives (59) .
For the nonlinear equation, put D t = ∣ X t − X t M , Λ ∣ D_t=\left|X_t-X_t^{M,\Lambda}\right| D t = X t − X t M , Λ L a , T L_{a,T} L a , T L σ , T L_{\sigma,T} L σ , T
S M : = sup t ≤ T ∣ σ ( X t M , Λ , t ) ∣ ≤ K T + L T ∥ X M , Λ ∥ ∞ ; [ 0 , T ] . S_M:=\sup_{t\le T}\left|\sigma\left(X_t^{M,\Lambda},t\right)\right|\le K_T+L_T\left\|X^{M,\Lambda}\right\|_{\infty;[0,T]}. S M := t ≤ T sup σ ( X t M , Λ , t ) ≤ K T + L T X M , Λ ∞ ; [ 0 , T ] . Subtracting the equations yields
D t ≤ L a , T ∫ 0 t D s d s + L σ , T ∫ 0 t ∣ V s ∣ D s d s D_t\le L_{a,T}\int_0^t D_s\,\mathrm{d}s+L_{\sigma,T}\int_0^t |V_s|D_s\,\mathrm{d}s D t ≤ L a , T ∫ 0 t D s d s + L σ , T ∫ 0 t ∣ V s ∣ D s d s + S M ∫ 0 t ∣ V s − Y s M , Λ ∣ d s . \qquad+S_M\int_0^t\left|V_s-\mathcal{Y}^{M,\Lambda}_s\right|\,\mathrm{d}s. + S M ∫ 0 t V s − Y s M , Λ d s . This decomposition leaves the exact velocity V V V D s D_s D s
Gronwall’s inequality with respect to the random finite measure L a , T d s + L σ , T ∣ V s ∣ d s L_{a,T}\,\mathrm{d}s+L_{\sigma,T}|V_s|\,\mathrm{d}s L a , T d s + L σ , T ∣ V s ∣ d s
sup t ≤ T D t ≤ T S M e L a , T T + L σ , T ∥ V ∥ L 1 ( 0 , T ) ∥ V − Y M , Λ ∥ L 2 ( 0 , T ) . \sup_{t\le T}D_t\le\sqrt{T}S_M e^{L_{a,T}T+L_{\sigma,T}\|V\|_{L^1(0,T)}}\left\|V-\mathcal{Y}^{M,\Lambda}\right\|_{L^2(0,T)}. t ≤ T sup D t ≤ T S M e L a , T T + L σ , T ∥ V ∥ L 1 ( 0 , T ) V − Y M , Λ L 2 ( 0 , T ) . By Proposition 3.4, E ∥ V ∥ L 2 ( 0 , T ) 2 < ∞ \mathbb{E}\|V\|_{L^2(0,T)}^2<\infty E ∥ V ∥ L 2 ( 0 , T ) 2 < ∞ g ∈ L 2 ( 0 , T ) g\in L^2(0,T) g ∈ L 2 ( 0 , T ) ∫ 0 T g ( t ) V t d t \int_0^T g(t)V_t\,\mathrm{d}t ∫ 0 T g ( t ) V t d t
K g ( λ , u ) : = b f ( u ) ∫ u T g ( t ) e − λ ( t − u ) d t . K_g(\lambda,u):=\sqrt{bf(u)}\int_u^T g(t)e^{-\lambda(t-u)}\,\mathrm{d}t. K g ( λ , u ) := b f ( u ) ∫ u T g ( t ) e − λ ( t − u ) d t . The Fubini condition follows from
∫ 0 T ∣ g ( t ) ∣ ( E ∣ V t ∣ 2 ) 1 / 2 d t ≤ ∥ g ∥ L 2 ( 0 , T ) ( E ∥ V ∥ L 2 ( 0 , T ) 2 ) 1 / 2 < ∞ . \int_0^T |g(t)|\left(\mathbb{E}|V_t|^2\right)^{1/2}\,\mathrm{d}t\le\|g\|_{L^2(0,T)}\left(\mathbb{E}\|V\|_{L^2(0,T)}^2\right)^{1/2}<\infty. ∫ 0 T ∣ g ( t ) ∣ ( E ∣ V t ∣ 2 ) 1/2 d t ≤ ∥ g ∥ L 2 ( 0 , T ) ( E ∥ V ∥ L 2 ( 0 , T ) 2 ) 1/2 < ∞. Hence V V V L 2 ( 0 , T ) L^2(0,T) L 2 ( 0 , T ) [11 Section 2.2.1] gives E exp ( c ∥ V ∥ L 2 ( 0 , T ) 2 ) < ∞ \mathbb{E}\exp(c\|V\|_{L^2(0,T)}^2)<\infty E exp ( c ∥ V ∥ L 2 ( 0 , T ) 2 ) < ∞ c > 0 c>0 c > 0 ∥ V ∥ L 1 ≤ T ∥ V ∥ L 2 \|V\|_{L^1}\le\sqrt{T}\|V\|_{L^2} ∥ V ∥ L 1 ≤ T ∥ V ∥ L 2 a y ≤ c y 2 + a 2 / ( 4 c ) ay\le cy^2+a^2/(4c) a y ≤ c y 2 + a 2 / ( 4 c ) a , y ≥ 0 a,y\ge0 a , y ≥ 0
It remains to control S M S_M S M
sup M ≥ 1 E ∥ Y M , Λ ∥ L 2 ( 0 , T ) 2 < ∞ , Λ = M 2 / 3 . \sup_{M\ge1}\mathbb{E}\left\|\mathcal{Y}^{M,\Lambda}\right\|_{L^2(0,T)}^2<\infty,\qquad\Lambda=M^{2/3}. M ≥ 1 sup E Y M , Λ L 2 ( 0 , T ) 2 < ∞ , Λ = M 2/3 . These velocities are centered Gaussian random elements of L 2 ( 0 , T ) L^2(0,T) L 2 ( 0 , T ) U U U C 0 > 0 C_0>0 C 0 > 0
E e ε ∥ U ∥ L 2 2 ≤ exp ( ε C 0 1 − 2 ε C 0 ) , 0 < ε < ( 2 C 0 ) − 1 . \mathbb{E}e^{\varepsilon\|U\|_{L^2}^2}\le\exp\left(\frac{\varepsilon C_0}{1-2\varepsilon C_0}\right),\qquad0<\varepsilon<(2C_0)^{-1}. E e ε ∥ U ∥ L 2 2 ≤ exp ( 1 − 2 ε C 0 ε C 0 ) , 0 < ε < ( 2 C 0 ) − 1 . Indeed, apply − log ( 1 − v ) ≤ v / ( 1 − v ) -\log(1-v)\le v/(1-v) − log ( 1 − v ) ≤ v / ( 1 − v ) ∥ Y M , Λ ∥ L 1 ( 0 , T ) \|\mathcal{Y}^{M,\Lambda}\|_{L^1(0,T)} ∥ Y M , Λ ∥ L 1 ( 0 , T )
1 + ∥ X M , Λ ∥ ∞ ; [ 0 , T ] ≤ ( 1 + ∣ x 0 ∣ ) exp ( C T [ T + ∥ Y M , Λ ∥ L 1 ( 0 , T ) ] ) . 1+\left\|X^{M,\Lambda}\right\|_{\infty;[0,T]}\le(1+|x_0|)\exp\left(C_T[T+\|\mathcal{Y}^{M,\Lambda}\|_{L^1(0,T)}]\right). 1 + X M , Λ ∞ ; [ 0 , T ] ≤ ( 1 + ∣ x 0 ∣ ) exp ( C T [ T + ∥ Y M , Λ ∥ L 1 ( 0 , T ) ] ) . Thus S M S_M S M S M S_M S M L p L^p L p □ \square □
The Young-regime rate
For 0 < b ≤ 1 0<b\le1 0 < b ≤ 1 1 / 2 1/2 1/2 1 / 2 < η < η ′ < H b ∘ 1/2<\eta<\eta'<H_b^\circ 1/2 < η < η ′ < H b ∘ η \eta η η ′ \eta' η ′ η ′ \eta' η ′ b = 1 b=1 b = 1 H b ∘ = 1 H_b^\circ=1 H b ∘ = 1
The kernel k λ k_\lambda k λ
∣ ∂ λ k λ ( r ) ∣ = ∣ ∫ 0 r v e − λ v d v ∣ ≤ r 2 2 . (61) |\partial_\lambda k_\lambda(r)|=\left|\int_0^r ve^{-\lambda v}\,\mathrm{d}v\right|\le\frac{r^2}{2}. \tag*{(61)} ∣ ∂ λ k λ ( r ) ∣ = ∫ 0 r v e − λ v d v ≤ 2 r 2 . (61) for all λ > 0 \lambda>0 λ > 0 r ≥ 0 r\ge0 r ≥ 0
For η ∈ ( 0 , 1 ) \eta\in(0,1) η ∈ ( 0 , 1 ) c η ( [ 0 , T ] ) c^\eta([0,T]) c η ([ 0 , T ]) C η ( [ 0 , T ] ) C^\eta([0,T]) C η ([ 0 , T ]) c η ( [ 0 , T ] ) c^\eta([0,T]) c η ([ 0 , T ]) η \eta η c η ( [ 0 , T ] ) c^\eta([0,T]) c η ([ 0 , T ])
Lemma 4.2 Let T > 0 T>0 T > 0 ( G m ) m ∈ I (G^m)_{m\in\mathcal{I}} ( G m ) m ∈ I [ 0 , T ] [0,T] [ 0 , T ] G 0 m = 0 G^m_0=0 G 0 m = 0 H ∈ ( 0 , 1 ) H\in(0,1) H ∈ ( 0 , 1 ) 0 ≤ K < ∞ 0\le K<\infty 0 ≤ K < ∞
sup m ∈ I E ∣ G t m − G s m ∣ 2 ≤ K ∣ t − s ∣ 2 H , s , t ∈ [ 0 , T ] . (62) \sup_{m\in\mathcal{I}}\mathbb{E}\left|G^m_t-G^m_s\right|^2\le K|t-s|^{2H},\qquad s,t\in[0,T]. \tag*{(62)} m ∈ I sup E ∣ G t m − G s m ∣ 2 ≤ K ∣ t − s ∣ 2 H , s , t ∈ [ 0 , T ] . (62) Then, for every η ∈ ( 0 , H ) \eta\in(0,H) η ∈ ( 0 , H ) c , C > 0 c,C>0 c , C > 0 K , H , η , T K,H,\eta,T K , H , η , T
sup m ∈ I E exp ( c ∥ G m ∥ C η ( [ 0 , T ] ) 2 ) ≤ C . (63) \sup_{m\in\mathcal{I}}\mathbb{E}\exp\left(c\left\|G^m\right\|_{C^\eta([0,T])}^2\right)\le C. \tag*{(63)} m ∈ I sup E exp ( c ∥ G m ∥ C η ([ 0 , T ]) 2 ) ≤ C . (63) Each G m G^m G m c η ( [ 0 , T ] ) c^\eta([0,T]) c η ([ 0 , T ])
Proof. The case K = 0 K=0 K = 0 K > 0 K>0 K > 0 t n , k = k T 2 − n t_{n,k}=kT2^{-n} t n , k = k T 2 − n Δ n , k G = G ( t n , k + 1 ) − G ( t n , k ) \Delta_{n,k}G=G(t_{n,k+1})-G(t_{n,k}) Δ n , k G = G ( t n , k + 1 ) − G ( t n , k ) x x x x ( 0 ) = 0 x(0)=0 x ( 0 ) = 0
∥ x ∥ C η ( [ 0 , T ] ) ≤ C η , T sup n ≥ 0 2 n η max 0 ≤ k < 2 n ∣ Δ n , k x ∣ . (64) \|x\|_{C^\eta([0,T])}\le C_{\eta,T}\sup_{n\ge0}2^{n\eta}\max_{0\le k<2^n}|\Delta_{n,k}x|. \tag*{(64)} ∥ x ∥ C η ([ 0 , T ]) ≤ C η , T n ≥ 0 sup 2 n η 0 ≤ k < 2 n max ∣ Δ n , k x ∣. (64) This follows by approximating the endpoints by their nested dyadic ancestors and summing the geometric series.
Each Δ n , k G m \Delta_{n,k}G^m Δ n , k G m K T 2 H 2 − 2 n H KT^{2H}2^{-2nH} K T 2 H 2 − 2 n H m m m
P ( sup n ≥ 0 2 n η max k < 2 n ∣ Δ n , k G m ∣ > x ) \mathbb{P}\left(\sup_{n\ge0}2^{n\eta}\max_{k<2^n}|\Delta_{n,k}G^m|>x\right) P ( n ≥ 0 sup 2 n η k < 2 n max ∣ Δ n , k G m ∣ > x ) ≤ 2 ∑ n = 0 ∞ 2 n exp ( − x 2 2 2 n ( H − η ) 2 K T 2 H ) . \le2 \sum_{n=0}^{\infty} 2^n \exp\left(-\frac{x^2 2^{2n(H-\eta)}}{2KT^{2H}}\right). ≤ 2 n = 0 ∑ ∞ 2 n exp ( − 2 K T 2 H x 2 2 2 n ( H − η ) ) . For x ≥ 1 x \ge1 x ≥ 1 C 1 e − c 1 x 2 C_1e^{-c_1x^2} C 1 e − c 1 x 2 m m m ∑ n ≥ 0 2 n e − c 2 2 n ( H − η ) \sum_{n\ge0}2^n e^{-c_2 2^{n(H-\eta)}} ∑ n ≥ 0 2 n e − c 2 2 n ( H − η ) c > 0 c>0 c > 0
Fix η 1 ∈ ( η , H ) \eta_1\in(\eta,H) η 1 ∈ ( η , H ) η 1 \eta_1 η 1 G m G^m G m η 1 \eta_1 η 1 G m , n G^{m,n} G m , n
∥ G m , n − G m ∥ C η ( [ 0 , T ] ) ≤ C η , η 1 , T 2 − n ( η 1 − η ) [ G m ] η 1 ; [ 0 , T ] . \left\|G^{m,n}-G^m\right\|_{C^\eta([0,T])}\le C_{\eta,\eta_1,T}2^{-n(\eta_1-\eta)}[G^m]_{\eta_1;[0,T]}. ∥ G m , n − G m ∥ C η ([ 0 , T ]) ≤ C η , η 1 , T 2 − n ( η 1 − η ) [ G m ] η 1 ; [ 0 , T ] . This follows by estimating increments separately below and above the mesh size. Each interpolant is a measurable function of finitely many Gaussian values and belongs to C η ( [ 0 , T ] ) C^\eta([0,T]) C η ([ 0 , T ])
Theorem 4.3 Let T > 0 T>0 T > 0 M ∈ N M\in\mathbb{N} M ∈ N Λ > 0 \Lambda>0 Λ > 0 0 < b ≤ 1 0<b\le1 0 < b ≤ 1 f ∈ L ∞ ( 0 , T ) f\in L^\infty(0,T) f ∈ L ∞ ( 0 , T ) [ 0 , Λ ] [0,\Lambda] [ 0 , Λ ] δ λ \delta_\lambda δ λ
sup t ≤ T E ∣ ζ t , f , b − ζ t M , Λ ∣ 2 ≤ b ∥ f ∥ ∞ , T T 5 20 ( 2 − b ) Γ ( 2 − b ) δ λ 2 Λ 2 − b + ∥ f ∥ ∞ , T T Γ ( 2 − b ) Λ − b . (65) \sup_{t\le T}\mathbb{E}\left|\zeta_{t,f,b}-\zeta_t^{M,\Lambda}\right|^2
\le\frac{b\|f\|_{\infty,T}T^5}{20(2-b)\Gamma(2-b)}\delta_\lambda^2\Lambda^{2-b}
+\frac{\|f\|_{\infty,T}T}{\Gamma(2-b)}\Lambda^{-b}. \tag*{(65)} t ≤ T sup E ζ t , f , b − ζ t M , Λ 2 ≤ 20 ( 2 − b ) Γ ( 2 − b ) b ∥ f ∥ ∞ , T T 5 δ λ 2 Λ 2 − b + Γ ( 2 − b ) ∥ f ∥ ∞ , T T Λ − b . (65) Set Λ = M 1 / 2 \Lambda=M^{1/2} Λ = M 1/2 δ λ ≤ C m e s h M − 1 / 2 \delta_\lambda\le C_{\mathrm{mesh}}M^{-1/2} δ λ ≤ C mesh M − 1/2
sup t ≤ T ∥ ζ t , f , b − ζ t M , Λ ∥ L 2 ( Ω ) ≤ C M − b / 4 . (66) \sup_{t\le T}\left\|\zeta_{t,f,b}-\zeta_t^{M,\Lambda}\right\|_{L^2(\Omega)}
\le CM^{-b/4}. \tag*{(66)} t ≤ T sup ζ t , f , b − ζ t M , Λ L 2 ( Ω ) ≤ C M − b /4 . (66) Moreover, for every p ≥ 2 p\ge2 p ≥ 2
( E sup t ≤ T ∣ ζ t , f , b − ζ t M , Λ ∣ p ) 1 / p ≤ C p M − b / 4 log ( 2 + M ) . (67) \left(\mathbb{E}\sup_{t\le T}\left|\zeta_{t,f,b}-\zeta_t^{M,\Lambda}\right|^p\right)^{1/p}
\le C_pM^{-b/4}\sqrt{\log(2+M)}. \tag*{(67)} ( E t ≤ T sup ζ t , f , b − ζ t M , Λ p ) 1/ p ≤ C p M − b /4 log ( 2 + M ) . (67) For every p ≥ 2 p\ge2 p ≥ 2 1 / 2 < η < η ′ < H b ∘ 1/2<\eta<\eta'<H_b^\circ 1/2 < η < η ′ < H b ∘
( E ∥ ζ f , b − ζ M , Λ ∥ C η ( [ 0 , T ] ) p ) 1 / p ≤ C p , η , η ′ ( M − b / 4 log ( 2 + M ) ) 1 − η / η ′ . (68) \left(\mathbb{E}\left\|\zeta_{f,b}-\zeta^{M,\Lambda}\right\|_{C^\eta([0,T])}^p\right)^{1/p}
\le C_{p,\eta,\eta'}\left(M^{-b/4}\sqrt{\log(2+M)}\right)^{1-\eta/\eta'}. \tag*{(68)} ( E ζ f , b − ζ M , Λ C η ([ 0 , T ]) p ) 1/ p ≤ C p , η , η ′ ( M − b /4 log ( 2 + M ) ) 1 − η / η ′ . (68) Proof Let e t M , Λ : = ζ t , f , b − ζ t M , Λ e_t^{M,\Lambda}:=\zeta_{t,f,b}-\zeta_t^{M,\Lambda} e t M , Λ := ζ t , f , b − ζ t M , Λ W W W
E ∣ e t M , Λ ∣ 2 = b ∫ 0 t f ( u ) ∑ k = 1 M ∫ I k ∣ k λ ( t − u ) − k ξ k ( t − u ) ∣ 2 μ b ( d λ ) d u + b ∫ 0 t f ( u ) ∫ Λ ∞ k λ ( t − u ) 2 μ b ( d λ ) d u . \mathbb{E}\left|e_t^{M,\Lambda}\right|^2
=b\int_0^t f(u)\sum_{k=1}^{M}\int_{I_k}\left|k_\lambda(t-u)-k_{\xi_k}(t-u)\right|^2\mu_b(\mathrm{d}\lambda)\,\mathrm{d}u
+b\int_0^t f(u)\int_\Lambda^\infty k_\lambda(t-u)^2\mu_b(\mathrm{d}\lambda)\,\mathrm{d}u. E e t M , Λ 2 = b ∫ 0 t f ( u ) k = 1 ∑ M ∫ I k ∣ k λ ( t − u ) − k ξ k ( t − u ) ∣ 2 μ b ( d λ ) d u + b ∫ 0 t f ( u ) ∫ Λ ∞ k λ ( t − u ) 2 μ b ( d λ ) d u . By (61), the first term is bounded by
b ∥ f ∥ ∞ , T δ λ 2 4 μ b ( [ 0 , Λ ] ) ∫ 0 t ( t − u ) 4 d u , \frac{b\|f\|_{\infty,T}\delta_\lambda^2}{4}\mu_b([0,\Lambda])\int_0^t(t-u)^4\,\mathrm{d}u, 4 b ∥ f ∥ ∞ , T δ λ 2 μ b ([ 0 , Λ ]) ∫ 0 t ( t − u ) 4 d u , which is the first term in (65). Since k λ ( r ) ≤ 1 / λ k_{\lambda}(r) \le1/\lambda k λ ( r ) ≤ 1/ λ
b ∥ f ∥ ∞ , T T ∫ Λ ∞ λ − 2 μ b ( d λ ) = ∥ f ∥ ∞ , T T Γ ( 2 − b ) Λ − b . b\|f\|_{\infty,T}T\int_{\Lambda}^{\infty}\lambda^{-2}\mu_b(\mathrm{d}\lambda)=\frac{\|f\|_{\infty,T}T}{\Gamma(2-b)}\Lambda^{-b}. b ∥ f ∥ ∞ , T T ∫ Λ ∞ λ − 2 μ b ( d λ ) = Γ ( 2 − b ) ∥ f ∥ ∞ , T T Λ − b . Under δ λ ≤ C m e s h Λ / M \delta_{\lambda}\le C_{\mathrm{mesh}}\Lambda/M δ λ ≤ C mesh Λ/ M M − 2 Λ 4 − b M^{-2}\Lambda^{4-b} M − 2 Λ 4 − b Λ − b \Lambda^{-b} Λ − b Λ = M 1 / 2 \Lambda=M^{1/2} Λ = M 1/2
More generally, (65) gives uniform pointwise mean-square convergence whenever Λ M → ∞ \Lambda M\to\infty Λ M → ∞ δ λ , M 2 Λ 2 − b → 0 \delta_{\lambda,M}^{2}\Lambda^{2-b}\to0 δ λ , M 2 Λ 2 − b → 0 M M M
For the uniform estimate, let
d M ( s , t ) 2 : = E ∣ e t M , Λ − e s M , Λ ∣ 2 , d_M(s,t)^2:=\mathbb{E}\left|e_t^{M,\Lambda}-e_s^{M,\Lambda}\right|^2, d M ( s , t ) 2 := E e t M , Λ − e s M , Λ 2 , and write h = t − s > 0 h=t-s>0 h = t − s > 0 b < 1 b<1 b < 1 C H , T h 2 H C_{H,T}h^{2H} C H , T h 2 H H ∈ ( 0 , 1 ) H\in(0,1) H ∈ ( 0 , 1 ) b = 1 b=1 b = 1 r = s − u r=s-u r = s − u
∣ ( k λ − k ξ ) ( r + h ) − ( k λ − k ξ ) ( r ) ∣ ≤ δ λ ∫ r r + h v d v ≤ δ λ h ( r + h ) , \left|(k_{\lambda}-k_{\xi})(r+h)-(k_{\lambda}-k_{\xi})(r)\right|
\le\delta_{\lambda}\int_r^{r+h}v\,\mathrm{d}v\le\delta_{\lambda}h(r+h), ∣ ( k λ − k ξ ) ( r + h ) − ( k λ − k ξ ) ( r ) ∣ ≤ δ λ ∫ r r + h v d v ≤ δ λ h ( r + h ) , and, on the interval ( s , t ] (s,t] ( s , t ] ∣ k λ ( v ) − k ξ ( v ) ∣ ≤ δ λ v 2 / 2 |k_{\lambda}(v)-k_{\xi}(v)|\le\delta_{\lambda}v^2/2 ∣ k λ ( v ) − k ξ ( v ) ∣ ≤ δ λ v 2 /2 u ∈ [ 0 , s ] u\in[0,s] u ∈ [ 0 , s ]
b ∥ f ∥ ∞ , T δ λ 2 μ b ( [ 0 , Λ ] ) h 2 ∫ 0 s ( s − u + h ) 2 d u ≤ C T δ λ 2 μ b ( [ 0 , Λ ] ) h 2 , b\|f\|_{\infty,T}\delta_{\lambda}^{2}\mu_b([0,\Lambda])h^2\int_0^s(s-u+h)^2\,\mathrm{d}u
\le C_T\delta_{\lambda}^{2}\mu_b([0,\Lambda])h^2, b ∥ f ∥ ∞ , T δ λ 2 μ b ([ 0 , Λ ]) h 2 ∫ 0 s ( s − u + h ) 2 d u ≤ C T δ λ 2 μ b ([ 0 , Λ ]) h 2 , and the contribution from u ∈ ( s , t ] u\in(s,t] u ∈ ( s , t ]
b ∥ f ∥ ∞ , T δ λ 2 4 μ b ( [ 0 , Λ ] ) ∫ s t ( t − u ) 4 d u = b ∥ f ∥ ∞ , T 20 δ λ 2 μ b ( [ 0 , Λ ] ) h 5 . \frac{b\|f\|_{\infty,T}\delta_{\lambda}^{2}}{4}\mu_b([0,\Lambda])\int_s^t(t-u)^4\,\mathrm{d}u
=\frac{b\|f\|_{\infty,T}}{20}\delta_{\lambda}^{2}\mu_b([0,\Lambda])h^5. 4 b ∥ f ∥ ∞ , T δ λ 2 μ b ([ 0 , Λ ]) ∫ s t ( t − u ) 4 d u = 20 b ∥ f ∥ ∞ , T δ λ 2 μ b ([ 0 , Λ ]) h 5 . For Λ = M 1 / 2 \Lambda=M^{1/2} Λ = M 1/2 δ λ ≤ C m e s h M − 1 / 2 \delta_{\lambda}\le C_{\mathrm{mesh}}M^{-1/2} δ λ ≤ C mesh M − 1/2
δ λ 2 μ b ( [ 0 , Λ ] ) ≤ C m e s h 2 M − b / 2 ( 2 − b ) Γ ( 2 − b ) ≤ C b , C m e s h . \delta_{\lambda}^{2}\mu_b([0,\Lambda])
\le\frac{C_{\mathrm{mesh}}^{2}M^{-b/2}}{(2-b)\Gamma(2-b)}
\le C_{b,C_{\mathrm{mesh}}}. δ λ 2 μ b ([ 0 , Λ ]) ≤ ( 2 − b ) Γ ( 2 − b ) C mesh 2 M − b /2 ≤ C b , C mesh . If b < 1 b<1 b < 1 h 2 ≤ T 1 − b h 1 + b h^2\le T^{1-b}h^{1+b} h 2 ≤ T 1 − b h 1 + b h 5 ≤ T 4 − b h 1 + b h^5\le T^{4-b}h^{1+b} h 5 ≤ T 4 − b h 1 + b
d M ( s , t ) 2 ≤ C ∣ t − s ∣ 1 + b (69) d_M(s,t)^2\le C|t-s|^{1+b} \tag*{(69)} d M ( s , t ) 2 ≤ C ∣ t − s ∣ 1 + b (69) uniformly in M M M b = 1 b=1 b = 1 H < 1 H<1 H < 1 h 2 + h 5 ≤ C H , T h 2 H h^2+h^5\le C_{H,T}h^{2H} h 2 + h 5 ≤ C H , T h 2 H
d M ( s , t ) 2 ≤ C H , T ∣ t − s ∣ 2 H . (70) d_M(s,t)^2\le C_{H,T}|t-s|^{2H}. \tag*{(70)} d M ( s , t ) 2 ≤ C H , T ∣ t − s ∣ 2 H . (70) By (66), the canonical diameter is O ( M − b / 4 ) O(M^{-b/4}) O ( M − b /4 ) H = ( 1 + b ) / 2 H=(1+b)/2 H = ( 1 + b ) /2 b < 1 b<1 b < 1 H ∈ ( 0 , 1 ) H\in(0,1) H ∈ ( 0 , 1 ) H > η ′ H>\eta' H > η ′ ε > 0 \varepsilon>0 ε > 0 N ( [ 0 , T ] , d M , ε ) N([0,T],d_M,\varepsilon) N ([ 0 , T ] , d M , ε ) d M d_M d M ε \varepsilon ε [ 0 , T ] [0,T] [ 0 , T ]
N ( [ 0 , T ] , d M , ε ) ≤ 1 + C ε − 1 / H . N([0,T],d_M,\varepsilon)\le1+C\varepsilon^{-1/H}. N ([ 0 , T ] , d M , ε ) ≤ 1 + C ε − 1/ H . Apply Dudley’s entropy bound [2 Theorem 1.3.3] to the centered Gaussian process ( t , κ ) ↦ κ t M , Λ (t,\kappa)\mapsto\kappa_t^{M,\Lambda} ( t , κ ) ↦ κ t M , Λ [ 0 , T ] × { − 1 , 1 } [0,T]\times\{-1,1\} [ 0 , T ] × { − 1 , 1 } 2 N ( [ 0 , T ] , d M , ε ) 2N([0,T],d_M,\varepsilon) 2 N ([ 0 , T ] , d M , ε ) a M : = C M − b / 4 a_M:=CM^{-b/4} a M := C M − b /4 C C C
E sup t ≤ T ∣ e t M , Λ ∣ ≤ C ∫ 0 a M log ( 1 + C ε − 1 / H ) d ε ≤ C M − b / 4 log ( 2 + M ) . \mathbb{E}\sup_{t\le T}\left|e_t^{M,\Lambda}\right|
\le C\int_0^{a_M}\sqrt{\log(1+C\varepsilon^{-1/H})}\,\mathrm{d}\varepsilon
\le CM^{-b/4}\sqrt{\log(2+M)}. E t ≤ T sup e t M , Λ ≤ C ∫ 0 a M log ( 1 + C ε − 1/ H ) d ε ≤ C M − b /4 log ( 2 + M ) . For the last inequality, substitute ε = a M u \varepsilon= a_M u ε = a M u C { log ( 2 + M ) + log ( 1 / u ) } C\{\log(2 + M) + \log(1/u)\} C { log ( 2 + M ) + log ( 1/ u )} ∫ 0 1 log ( 1 / u ) d u < ∞ \int_0^1 \sqrt{\log(1/u)}\,\mathrm{d}u < \infty ∫ 0 1 log ( 1/ u ) d u < ∞ sup t ≤ T ∥ e t M , Λ ∥ L 2 ( Ω ) ≤ C M − b / 4 \sup_{t\le T}\|e_t^{M,\Lambda}\|_{L^2(\Omega)} \le C M^{-b/4} sup t ≤ T ∥ e t M , Λ ∥ L 2 ( Ω ) ≤ C M − b /4 [2 Theorem 2.1.1] yields, for every p ≥ 2 p \ge2 p ≥ 2
∥ sup t ≤ T ∣ e t M , Λ ∣ ∥ L p ( Ω ) ≤ E sup t ≤ T ∣ e t M , Λ ∣ + C p M − b / 4 , \left\|\sup_{t\le T}|e_t^{M,\Lambda}|\right\|_{L^p(\Omega)} \le\mathbb{E}\sup_{t\le T}|e_t^{M,\Lambda}| + C_pM^{-b/4}, t ≤ T sup ∣ e t M , Λ ∣ L p ( Ω ) ≤ E t ≤ T sup ∣ e t M , Λ ∣ + C p M − b /4 , after integrating its Gaussian tail bound. This proves (67) .
Finally, the increment bounds and Lemma 4.2 give exponential moments of ∥ e M , Λ ∥ C η ′ ( [ 0 , T ] ) \|e^{M,\Lambda}\|_{C^{\eta'}([0,T])} ∥ e M , Λ ∥ C η ′ ([ 0 , T ]) M M M η ′ < H b ∘ \eta' < H_b^\circ η ′ < H b ∘
∣ x t − x s ∣ ≤ min { 2 ∥ x ∥ ∞ ; [ 0 , T ] , [ x ] η ′ ; [ 0 , T ] ∣ t − s ∣ η ′ } . |x_t-x_s| \le\min\{2\|x\|_{\infty;[0,T]},[x]_{\eta';[0,T]}|t-s|^{\eta'}\}. ∣ x t − x s ∣ ≤ min { 2∥ x ∥ ∞ ; [ 0 , T ] , [ x ] η ′ ; [ 0 , T ] ∣ t − s ∣ η ′ } . Interpolating between these bounds, and putting ϑ : = η / η ′ ∈ ( 0 , 1 ) \vartheta:= \eta/\eta' \in(0,1) ϑ := η / η ′ ∈ ( 0 , 1 )
∥ x ∥ C η ( [ 0 , T ] ) ≤ C ∥ x ∥ ∞ ; [ 0 , T ] 1 − ϑ ∥ x ∥ C η ′ ( [ 0 , T ] ) ϑ . \|x\|_{C^\eta([0,T])} \le C\|x\|_{\infty;[0,T]}^{1-\vartheta}\|x\|_{C^{\eta'}([0,T])}^{\vartheta}. ∥ x ∥ C η ([ 0 , T ]) ≤ C ∥ x ∥ ∞ ; [ 0 , T ] 1 − ϑ ∥ x ∥ C η ′ ([ 0 , T ]) ϑ . Consequently, Hölder’s inequality yields
∥ ∥ e M , Λ ∥ C η ∥ L p ( Ω ) ≤ C ∥ ∥ e M , Λ ∥ ∞ ∥ L p ( Ω ) 1 − ϑ ∥ ∥ e M , Λ ∥ C η ′ ∥ L p ( Ω ) ϑ ≤ C p , η , η ′ ( M − b / 4 log ( 2 + M ) ) 1 − ϑ . \left\|\|e^{M,\Lambda}\|_{C^\eta}\right\|_{L^p(\Omega)}
\le C\left\|\|e^{M,\Lambda}\|_{\infty}\right\|_{L^p(\Omega)}^{1-\vartheta}
\left\|\|e^{M,\Lambda}\|_{C^{\eta'}}\right\|_{L^p(\Omega)}^\vartheta
\le C_{p,\eta,\eta'}\left(M^{-b/4}\sqrt{\log(2+M)}\right)^{1-\vartheta}. ∥ e M , Λ ∥ C η L p ( Ω ) ≤ C ∥ e M , Λ ∥ ∞ L p ( Ω ) 1 − ϑ ∥ e M , Λ ∥ C η ′ L p ( Ω ) ϑ ≤ C p , η , η ′ ( M − b /4 log ( 2 + M ) ) 1 − ϑ . Here all path norms are taken on [ 0 , T ] [0,T] [ 0 , T ] (68) . The factor 1 − η / η ′ 1-\eta/\eta' 1 − η / η ′
Assumption 4.4 The coefficients a , σ a,\sigma a , σ σ \sigma σ x x x T > 0 T>0 T > 0 0 ≤ K T , L T < ∞ 0 \le K_T,L_T < \infty 0 ≤ K T , L T < ∞
∣ a ( 0 , t ) ∣ ≤ K T , |a(0,t)| \le K_T, ∣ a ( 0 , t ) ∣ ≤ K T , ∣ a ( x , t ) − a ( y , t ) ∣ ≤ L T ∣ x − y ∣ , |a(x,t)-a(y,t)| \le L_T|x-y|, ∣ a ( x , t ) − a ( y , t ) ∣ ≤ L T ∣ x − y ∣ , ∣ σ ( x , t ) ∣ + ∣ ∂ x σ ( x , t ) ∣ ≤ K T , |\sigma(x,t)|+|\partial_x\sigma(x,t)| \le K_T, ∣ σ ( x , t ) ∣ + ∣ ∂ x σ ( x , t ) ∣ ≤ K T , ∣ σ ( x , t ) − σ ( y , s ) ∣ ≤ L T ( ∣ x − y ∣ + ∣ t − s ∣ ) , |\sigma(x,t)-\sigma(y,s)| \le L_T(|x-y|+|t-s|), ∣ σ ( x , t ) − σ ( y , s ) ∣ ≤ L T ( ∣ x − y ∣ + ∣ t − s ∣ ) , ∣ ∂ x σ ( x , t ) − ∂ x σ ( y , s ) ∣ ≤ L T ( ∣ x − y ∣ + ∣ t − s ∣ ) , |\partial_x\sigma(x,t)-\partial_x\sigma(y,s)| \le L_T(|x-y|+|t-s|), ∣ ∂ x σ ( x , t ) − ∂ x σ ( y , s ) ∣ ≤ L T ( ∣ x − y ∣ + ∣ t − s ∣ ) , x , y ∈ R x,y\in\mathbb{R} x , y ∈ R s , t ∈ [ 0 , T ] s,t\in[0,T] s , t ∈ [ 0 , T ]
Assumption 4.4 implies Assumption 3.5 with α = 1 \alpha=1 α = 1
Lemma 4.5 Let T > 0 T>0 T > 0 η ∈ ( 1 / 2 , 1 ) \eta\in(1/2,1) η ∈ ( 1/2 , 1 ) z , z ~ ∈ C η ( [ 0 , T ] ) z,\tilde z\in C^\eta([0,T]) z , z ~ ∈ C η ([ 0 , T ]) z ( 0 ) = z ~ ( 0 ) = 0 z(0)=\tilde z(0)=0 z ( 0 ) = z ~ ( 0 ) = 0 x x x x ~ \tilde x x ~
x t = x 0 + ∫ 0 t a ( x s , s ) d s + ∫ 0 t σ ( x s , s ) d z s x_t=x_0+\int_0^t a(x_s,s)\,\mathrm{d}s+\int_0^t \sigma(x_s,s)\,\mathrm{d}z_s x t = x 0 + ∫ 0 t a ( x s , s ) d s + ∫ 0 t σ ( x s , s ) d z s and the corresponding equation with initial value x ~ 0 \tilde x_0 x ~ 0 z ~ \tilde z z ~ q : = 1 / η q:=1/\eta q := 1/ η
∥ x − x ~ ∥ ∞ ; [ 0 , T ] ≤ C exp ( C [ 1 + ∥ z ∥ C η ( [ 0 , T ] ) q + ∥ z ~ ∥ C η ( [ 0 , T ] ) q ] ) ( ∣ x 0 − x ~ 0 ∣ + ∥ z − z ~ ∥ C η ( [ 0 , T ] ) ) . (71) \|x-\tilde x\|_{\infty;[0,T]}
\le C\exp\left(C\left[1+\|z\|_{C^\eta([0,T])}^q+\|\tilde z\|_{C^\eta([0,T])}^q\right]\right)
\left(|x_0-\tilde x_0|+\|z-\tilde z\|_{C^\eta([0,T])}\right). \tag*{(71)} ∥ x − x ~ ∥ ∞ ; [ 0 , T ] ≤ C exp ( C [ 1 + ∥ z ∥ C η ([ 0 , T ]) q + ∥ z ~ ∥ C η ([ 0 , T ]) q ] ) ( ∣ x 0 − x ~ 0 ∣ + ∥ z − z ~ ∥ C η ([ 0 , T ]) ) . (71) where C C C T T T η \eta η A T = sup t ≤ T ∣ a ( 0 , t ) ∣ A_T=\sup_{t\leq T}|a(0,t)| A T = sup t ≤ T ∣ a ( 0 , t ) ∣ a a a σ \sigma σ ∂ x σ \partial_x\sigma ∂ x σ ∣ x 0 ∣ + ∣ x ~ 0 ∣ |x_0|+|\widetilde{x}_0| ∣ x 0 ∣ + ∣ x 0 ∣ R × [ 0 , T ] \mathbb{R}\times[0,T] R × [ 0 , T ] C C C sup ( x , t ) ∈ R × [ 0 , T ] ∣ a ( x , t ) ∣ \sup_{(x,t)\in\mathbb{R}\times[0,T]}|a(x,t)| sup ( x , t ) ∈ R × [ 0 , T ] ∣ a ( x , t ) ∣
The proof uses a partition on which the local stability terms can be absorbed. Its number of intervals is of order at most 1 + [ ∥ z ∥ η ; [ 0 , T ] ] 1 / η + [ ∥ z ~ ∥ η ; [ 0 , T ] ] 1 / η 1+[\lVert z\rVert_{\eta;[0,T]}]^{1/\eta}+[\lVert\widetilde{z}\rVert_{\eta;[0,T]}]^{1/\eta} 1 + [∥ z ∥ η ; [ 0 , T ] ] 1/ η + [∥ z ∥ η ; [ 0 , T ] ] 1/ η
Proof Step 1: A priori bounds. Fix p 0 ∈ ( 1 / η , 2 ) p_0\in(1/\eta,2) p 0 ∈ ( 1/ η , 2 ) η \eta η α = 1 \alpha=1 α = 1 p 0 p_0 p 0 [9 Lemma 2.4, Eq. (2.8)] and the Lipschitz bounds on σ \sigma σ I = [ a , c ] I=[a,c] I = [ a , c ] Δ ≤ 1 \Delta\leq1 Δ ≤ 1 K : = ∥ z ∥ η ; [ 0 , T ] K:=\lVert z\rVert_{\eta;[0,T]} K := ∥ z ∥ η ; [ 0 , T ]
∥ x ∥ p 0 -var ; I ≤ C Δ ( 1 + ∣ x a ∣ ) + C ( Δ + K Δ η ) ∥ x ∥ p 0 -var ; I + C K Δ η . \lVert x\rVert_{p_0\text{-var};I}\leq C\Delta(1+|x_a|)+C(\Delta+K\Delta^\eta)\lVert x\rVert_{p_0\text{-var};I}+CK\Delta^\eta. ∥ x ∥ p 0 -var ; I ≤ C Δ ( 1 + ∣ x a ∣ ) + C ( Δ + K Δ η ) ∥ x ∥ p 0 -var ; I + C K Δ η . Here the linear-growth drift contributes at most C Δ ( 1 + ∣ x a ∣ + ∥ x ∥ p 0 -var ; I ) C\Delta(1+|x_a|+\lVert x\rVert_{p_0\text{-var};I}) C Δ ( 1 + ∣ x a ∣ + ∥ x ∥ p 0 -var ; I ) ∥ z ∥ p 0 -var ; I ≤ [ z ] η ; [ 0 , T ] Δ η \lVert z\rVert_{p_0\text{-var};I}\leq[z]_{\eta;[0,T]}\Delta^\eta ∥ z ∥ p 0 -var ; I ≤ [ z ] η ; [ 0 , T ] Δ η ∥ σ ( x ⋅ , ⋅ ) ∥ p 0 -var ; I ≤ C ( ∥ x ∥ p 0 -var ; I + Δ ) \lVert\sigma(x_\cdot,\cdot)\rVert_{p_0\text{-var};I}\leq C(\lVert x\rVert_{p_0\text{-var};I}+\Delta) ∥ σ ( x ⋅ , ⋅ ) ∥ p 0 -var ; I ≤ C (∥ x ∥ p 0 -var ; I + Δ ) Δ ≤ c ∗ ( 1 + K ) − 1 / η \Delta\leq c_*(1+K)^{-1/\eta} Δ ≤ c ∗ ( 1 + K ) − 1/ η c ∗ > 0 c_*>0 c ∗ > 0
∥ x ∥ p 0 -var ; I ≤ C Δ ( 1 + ∣ x a ∣ ) + C K Δ η . \lVert x\rVert_{p_0\text{-var};I}\leq C\Delta(1+|x_a|)+CK\Delta^\eta. ∥ x ∥ p 0 -var ; I ≤ C Δ ( 1 + ∣ x a ∣ ) + C K Δ η . On an equal partition of mesh h h h min { T , 1 , c ∗ ( 1 + K ) − 1 / η } \min\{T,1,c_*(1+K)^{-1/\eta}\} min { T , 1 , c ∗ ( 1 + K ) − 1/ η }
1 + ∣ x t j + 1 ∣ ≤ ( 1 + C h ) ( 1 + ∣ x t j ∣ ) + C K h η . 1+|x_{t_{j+1}}|\leq(1+Ch)(1+|x_{t_j}|)+CKh^\eta. 1 + ∣ x t j + 1 ∣ ≤ ( 1 + C h ) ( 1 + ∣ x t j ∣ ) + C K h η . Discrete Gronwall, using ∏ j ( 1 + C h ) ≤ e C T \prod_j(1+Ch)\leq e^{CT} ∏ j ( 1 + C h ) ≤ e C T ∑ j K h η ≤ C T ( 1 + K 1 / η ) \sum_j Kh^\eta\leq C_T(1+K^{1/\eta}) ∑ j K h η ≤ C T ( 1 + K 1/ η )
∥ x ∥ ∞ ; [ 0 , T ] ≤ C T ( 1 + ∣ x 0 ∣ + K 1 / η ) . \lVert x\rVert_{\infty;[0,T]}\leq C_T(1+|x_0|+K^{1/\eta}). ∥ x ∥ ∞ ; [ 0 , T ] ≤ C T ( 1 + ∣ x 0 ∣ + K 1/ η ) . The same bound holds for x ~ \widetilde{x} x
R : = ( 1 + ∣ x 0 ∣ + ∣ x ~ 0 ∣ + ( 1 + [ z ] η ; [ 0 , T ] + [ z ~ ] η ; [ 0 , T ] ) 1 / η ) η . R:=\left(1+|x_0|+|\widetilde{x}_0|+\left(1+[z]_{\eta;[0,T]}+[\widetilde{z}]_{\eta;[0,T]}\right)^{1/\eta}\right)^\eta. R := ( 1 + ∣ x 0 ∣ + ∣ x 0 ∣ + ( 1 + [ z ] η ; [ 0 , T ] + [ z ] η ; [ 0 , T ] ) 1/ η ) η . Both solution norms are bounded by C R 1 / η CR^{1/\eta} C R 1/ η K ≤ R K\leq R K ≤ R I I I Δ ≤ min { 1 , c ∗ R − 1 / η } \Delta\leq\min\{1,c_*R^{-1/\eta}\} Δ ≤ min { 1 , c ∗ R − 1/ η } [ u , v ] ⊂ I [u,v]\subset I [ u , v ] ⊂ I
∣ x v − x u ∣ ≤ C R 1 / η ∣ v − u ∣ + C R ∣ v − u ∣ η . |x_v-x_u|\leq CR^{1/\eta}|v-u|+CR|v-u|^\eta. ∣ x v − x u ∣ ≤ C R 1/ η ∣ v − u ∣ + C R ∣ v − u ∣ η . The same estimate holds for x ~ \widetilde{x} x ∣ v − u ∣ η |v-u|^\eta ∣ v − u ∣ η
Δ η ( [ x ] η ; I + [ x ~ ] η ; I ) ≤ C ( R 1 / η Δ + R Δ η ) ≤ C R Δ η . (72) \Delta^\eta\left([x]_{\eta;I}+[\widetilde{x}]_{\eta;I}\right)\leq C(R^{1/\eta}\Delta+R\Delta^\eta)\leq CR\Delta^\eta. \tag*{(72)} Δ η ( [ x ] η ; I + [ x ] η ; I ) ≤ C ( R 1/ η Δ + R Δ η ) ≤ C R Δ η . (72) The last inequality follows from R 1 / η − 1 Δ 1 − η = ( R 1 / η Δ ) 1 − η ≤ c ∗ 1 − η R^{1/\eta-1}\Delta^{1-\eta}=(R^{1/\eta}\Delta)^{1-\eta}\leq c_*^{1-\eta} R 1/ η − 1 Δ 1 − η = ( R 1/ η Δ ) 1 − η ≤ c ∗ 1 − η
Step 2: Local difference estimate. Write e = x − x ~ e=x-\widetilde{x} e = x − x h = z − z ~ h=z-\widetilde{z} h = z − z
∥ v ∥ η , I ∗ : = ∥ v ∥ ∞ ; I + Δ η [ v ] η ; I . \lVert v\rVert_{\eta,I}^*:=\lVert v\rVert_{\infty;I}+\Delta^\eta[v]_{\eta;I}. ∥ v ∥ η , I ∗ := ∥ v ∥ ∞ ; I + Δ η [ v ] η ; I . The Young–Loeve inequality [9 Eq. (2.6)] gives
∣ ∫ s t u r d v r − u s ( v t − v s ) ∣ ≤ C η [ u ] η ; I [ v ] η ; I ∣ t − s ∣ 2 η , a ≤ s < t ≤ c . \left|\int_s^t u_r\,dv_r-u_s(v_t-v_s)\right|\leq C_\eta[u]_{\eta;I}[v]_{\eta;I}|t-s|^{2\eta},\qquad a\leq s<t\leq c. ∫ s t u r d v r − u s ( v t − v s ) ≤ C η [ u ] η ; I [ v ] η ; I ∣ t − s ∣ 2 η , a ≤ s < t ≤ c . Consequently, for J ( t ) = ∫ a t u r d v r J(t)=\int_a^t u_r\,dv_r J ( t ) = ∫ a t u r d v r
∥ J ∥ η , I ∗ ≤ C η [ v ] η ; I Δ η ( ∥ u ∥ ∞ ; I + Δ η [ u ] η ; I ) . \lVert J\rVert_{\eta,I}^*\leq C_\eta[v]_{\eta;I}\Delta^\eta\left(\lVert u\rVert_{\infty;I}+\Delta^\eta[u]_{\eta;I}\right). ∥ J ∥ η , I ∗ ≤ C η [ v ] η ; I Δ η ( ∥ u ∥ ∞ ; I + Δ η [ u ] η ; I ) . Split the difference of the Young integrals into
∫ a t ( σ ( x r , r ) − σ ( x ~ r , r ) ) d z r + ∫ a t σ ( x ~ r , r ) d h r . \int_a^t \bigl(\sigma(x_r,r)-\sigma(\widetilde{x}_r,r)\bigr)\,dz_r+\int_a^t \sigma(\widetilde{x}_r,r)\,dh_r. ∫ a t ( σ ( x r , r ) − σ ( x r , r ) ) d z r + ∫ a t σ ( x r , r ) d h r . To estimate the first integrand, write
σ ( x r , r ) − σ ( x ~ r , r ) = A r e r , A r : = ∫ 0 1 ∂ x σ ( ( 1 − θ ) x ~ r + θ x r , r ) d θ . \sigma(x_r,r)-\sigma(\widetilde{x}_r,r)=A_r e_r,\qquad A_r:=\int_0^1\partial_x\sigma\bigl((1-\theta)\widetilde{x}_r+\theta x_r,r\bigr)\,d\theta. σ ( x r , r ) − σ ( x r , r ) = A r e r , A r := ∫ 0 1 ∂ x σ ( ( 1 − θ ) x r + θ x r , r ) d θ . The coefficient assumptions imply ∥ A ∥ ∞ ; I ≤ C \lVert A\rVert_{\infty;I}\le C ∥ A ∥ ∞ ; I ≤ C
[ A ] η ; I ≤ C ( [ x ] η ; I + [ x ~ ] η ; I + Δ 1 − η ) . [A]_{\eta;I}\le C\bigl([x]_{\eta;I}+[\widetilde{x}]_{\eta;I}+\Delta^{1-\eta}\bigr). [ A ] η ; I ≤ C ( [ x ] η ; I + [ x ] η ; I + Δ 1 − η ) . The product increment identity for A e Ae A e
[ σ ( x ⋅ , ⋅ ) − σ ( x ~ ⋅ , ⋅ ) ] η ; I ≤ C [ e ] η ; I + C ∥ e ∥ ∞ ; I ( [ x ] η ; I + [ x ~ ] η ; I + Δ 1 − η ) . [\sigma(x_{\cdot},\cdot)-\sigma(\widetilde{x}_{\cdot},\cdot)]_{\eta;I}
\le C[e]_{\eta;I}+C\lVert e\rVert_{\infty;I}\bigl([x]_{\eta;I}+[\widetilde{x}]_{\eta;I}+\Delta^{1-\eta}\bigr). [ σ ( x ⋅ , ⋅ ) − σ ( x ⋅ , ⋅ ) ] η ; I ≤ C [ e ] η ; I + C ∥ e ∥ ∞ ; I ( [ x ] η ; I + [ x ] η ; I + Δ 1 − η ) . Together with (72), this bounds the first Young integral in the scaled norm by
C ( R Δ η + ( R Δ η ) 2 ) ∥ e ∥ η , I ∗ . C\bigl(R\Delta^\eta+(R\Delta^\eta)^2\bigr)\lVert e\rVert^*_{\eta,I}. C ( R Δ η + ( R Δ η ) 2 ) ∥ e ∥ η , I ∗ . The second integral is bounded by
C Δ η ( 1 + R Δ η ) ∥ h ∥ C η ( [ 0 , T ] ) . C\Delta^\eta(1+R\Delta^\eta)\lVert h\rVert_{C^\eta([0,T])}. C Δ η ( 1 + R Δ η ) ∥ h ∥ C η ([ 0 , T ]) . Finally, spatial Lipschitz continuity of the drift bounds the scaled norm of its difference integral by C Δ ∥ e ∥ η , I ∗ C\Delta\lVert e\rVert^*_{\eta,I} C Δ ∥ e ∥ η , I ∗
∥ e ∥ η , I ∗ ≤ ∣ e a ∣ + C Δ η ( 1 + R Δ η ) ∥ h ∥ C η ( [ 0 , T ] ) + C ( Δ + R Δ η + ( R Δ η ) 2 ) ∥ e ∥ η , I ∗ . (73) \lVert e\rVert^*_{\eta,I}\le|e_a|+C\Delta^\eta(1+R\Delta^\eta)\lVert h\rVert_{C^\eta([0,T])}
+C\bigl(\Delta+R\Delta^\eta+(R\Delta^\eta)^2\bigr)\lVert e\rVert^*_{\eta,I}. \tag*{(73)} ∥ e ∥ η , I ∗ ≤ ∣ e a ∣ + C Δ η ( 1 + R Δ η ) ∥ h ∥ C η ([ 0 , T ]) + C ( Δ + R Δ η + ( R Δ η ) 2 ) ∥ e ∥ η , I ∗ . (73) Choose 0 < c ∗ ≤ 1 0<c_*\le1 0 < c ∗ ≤ 1 η \eta η C ( c ∗ + c ∗ η + c ∗ 2 η ) ≤ 1 / 2 C(c_*+c_*^\eta+c_*^{2\eta})\le1/2 C ( c ∗ + c ∗ η + c ∗ 2 η ) ≤ 1/2 Δ ≤ min ( 1 , c ∗ R − 1 / η ) \Delta\le\min(1,c_*R^{-1/\eta}) Δ ≤ min ( 1 , c ∗ R − 1/ η ) R Δ η ≤ c ∗ η R\Delta^\eta\le c_*^\eta R Δ η ≤ c ∗ η d : = min ( 1 , T , c ∗ R − 1 / η ) d:=\min(1,T,c_*R^{-1/\eta}) d := min ( 1 , T , c ∗ R − 1/ η ) n : = ⌈ T / d ⌉ n:=\lceil T/d\rceil n := ⌈ T / d ⌉ [ 0 , T ] [0,T] [ 0 , T ] n n n
n ≤ C T ( 1 + R 1 / η ) , n\le C_T(1+R^{1/\eta}), n ≤ C T ( 1 + R 1/ η ) , and the absorbed estimate on each interval gives
∥ e ∥ ∞ ; [ t j , t j + 1 ] ≤ C ( ∣ e t j ∣ + ∥ h ∥ C η ( [ 0 , T ] ) ) , j = 0 , … , n − 1 , \lVert e\rVert_{\infty;[t_j,t_{j+1}]}\le C\bigl(|e_{t_j}|+\lVert h\rVert_{C^\eta([0,T])}\bigr),\qquad j=0,\ldots,n-1, ∥ e ∥ ∞ ; [ t j , t j + 1 ] ≤ C ( ∣ e t j ∣ + ∥ h ∥ C η ([ 0 , T ]) ) , j = 0 , … , n − 1 , with C C C R R R j j j
∥ e ∥ ∞ ; [ 0 , T ] ≤ C n ( 1 + n ) ( ∣ e 0 ∣ + ∥ h ∥ C η ( [ 0 , T ] ) ) . \lVert e\rVert_{\infty;[0,T]}\le C^n(1+n)\bigl(|e_0|+\lVert h\rVert_{C^\eta([0,T])}\bigr). ∥ e ∥ ∞ ; [ 0 , T ] ≤ C n ( 1 + n ) ( ∣ e 0 ∣ + ∥ h ∥ C η ([ 0 , T ]) ) . Since q = 1 / η q=1/\eta q = 1/ η R q ≤ C q ( 1 + ∣ x 0 ∣ + ∣ x ~ 0 ∣ + ∥ z ∥ C η + ∥ z ~ ∥ C η ) R^q\le C_q(1+|x_0|+|\widetilde{x}_0|+\lVert z\rVert_{C^\eta}+\lVert\widetilde{z}\rVert_{C^\eta}) R q ≤ C q ( 1 + ∣ x 0 ∣ + ∣ x 0 ∣ + ∥ z ∥ C η + ∥ z ∥ C η ) C C C
If the drift is bounded, its variation contribution in the first estimate is C Δ C\Delta C Δ x a x_a x a ∥ x ∥ p 0 -var ; I ≤ C ( Δ + K Δ η ) \lVert x\rVert_{p_0\text{-var};I}\le C(\Delta+K\Delta^\eta) ∥ x ∥ p 0 -var ; I ≤ C ( Δ + K Δ η ) R = 1 + [ z ] η ; [ 0 , T ] + [ z ~ ] η ; [ 0 , T ] R=1+[z]_{\eta;[0,T]}+[\widetilde{z}]_{\eta;[0,T]} R = 1 + [ z ] η ; [ 0 , T ] + [ z ] η ; [ 0 , T ]
Theorem 4.6 Let T > 0 T>0 T > 0 M ∈ N M\in\mathbb{N} M ∈ N Λ = M 1 / 2 \Lambda=M^{1/2} Λ = M 1/2 0 < b ≤ 1 0<b\le1 0 < b ≤ 1 f ∈ L ∞ ( 0 , T ) f\in L^\infty(0,T) f ∈ L ∞ ( 0 , T ) δ λ ≤ C m e s h M − 1 / 2 \delta_\lambda\le C_{\mathrm{mesh}}M^{-1/2} δ λ ≤ C mesh M − 1/2 ξ k ∈ I k \xi_k\in I_k ξ k ∈ I k X X X X M , Λ X^{M,\Lambda} X M , Λ x 0 x_0 x 0 W W W p ≥ 2 p\ge2 p ≥ 2 1 / 2 < η < η ′ < H b ∘ 1/2<\eta<\eta'<H_b^\circ 1/2 < η < η ′ < H b ∘
( E sup t ≤ T ∣ X t − X t M , Λ ∣ p ) 1 / p ≤ C ( M − b / 4 log ( 2 + M ) ) 1 − η / η ′ . (74) \left(\mathbb{E}\sup_{t\le T}|X_t-X_t^{M,\Lambda}|^p\right)^{1/p}
\le C\left(M^{-b/4}\sqrt{\log(2+M)}\right)^{1-\eta/\eta'}. \tag*{(74)} ( E t ≤ T sup ∣ X t − X t M , Λ ∣ p ) 1/ p ≤ C ( M − b /4 log ( 2 + M ) ) 1 − η / η ′ . (74) The constant may depend on x 0 x_0 x 0 M M M
Proof. Apply Lemma 4.5 with q = 1 / η ∈ ( 1 , 2 ) q=1/\eta\in(1,2) q = 1/ η ∈ ( 1 , 2 ) z = ζ f , b z=\zeta_{f,b} z = ζ f , b z ~ = ζ M , Λ \widetilde{z}=\zeta^{M,\Lambda} z = ζ M , Λ Propositions 3.1 and 3.2 , together with (69) –(70) and Lemma 4.2 , give c 0 > 0 c_0>0 c 0 > 0
E e c 0 ∥ ζ f , b ∥ C η ( [ 0 , T ] ) 2 < ∞ , sup M ≥ 1 E e c 0 ∥ ζ f , b − ζ M , Λ ∥ C η ( [ 0 , T ] ) 2 < ∞ . (75) \mathbb{E}e^{c_0\|\zeta_{f,b}\|_{\mathcal{C}^{\eta}([0,T])}^{2}}<\infty,\qquad\sup_{M\geq1}\mathbb{E}e^{c_0\|\zeta_{f,b}-\zeta^{M,\Lambda}\|_{\mathcal{C}^{\eta}([0,T])}^{2}}<\infty. \tag*{(75)} E e c 0 ∥ ζ f , b ∥ C η ([ 0 , T ]) 2 < ∞ , M ≥ 1 sup E e c 0 ∥ ζ f , b − ζ M , Λ ∥ C η ([ 0 , T ]) 2 < ∞. (75) Since ζ M , Λ = ζ f , b − ( ζ f , b − ζ M , Λ ) \zeta^{M,\Lambda}=\zeta_{f,b}-(\zeta_{f,b}-\zeta^{M,\Lambda}) ζ M , Λ = ζ f , b − ( ζ f , b − ζ M , Λ ) c 0 c_0 c 0
sup M ≥ 1 E e c 0 ∥ ζ M , Λ ∥ C η ( [ 0 , T ] ) 2 < ∞ . \sup_{M\geq1}\mathbb{E}e^{c_0\|\zeta^{M,\Lambda}\|_{\mathcal{C}^{\eta}([0,T])}^{2}}<\infty. M ≥ 1 sup E e c 0 ∥ ζ M , Λ ∥ C η ([ 0 , T ]) 2 < ∞. For q < 2 q<2 q < 2 1 ≤ r < ∞ 1\leq r<\infty 1 ≤ r < ∞ x q ≤ ϵ x 2 + C q , ϵ x^q\leq\epsilon x^2+C_{q,\epsilon} x q ≤ ϵ x 2 + C q , ϵ ϵ \epsilon ϵ r r r c 0 c_0 c 0 r r r M M M
Denote this factor by K M K_M K M (71) and Hölder’s inequality give
∥ ∥ X − X M , Λ ∥ ∞ ∥ L p ( Ω ) ≤ ∥ K M ∥ L 2 p ( Ω ) ∥ ∥ ζ f , b − ζ M , Λ ∥ C η ∥ L 2 p ( Ω ) . \left\|\left\|X-X^{M,\Lambda}\right\|_{\infty}\right\|_{L^p(\Omega)}
\leq\|K_M\|_{L^{2p}(\Omega)}
\left\|\left\|\zeta_{f,b}-\zeta^{M,\Lambda}\right\|_{\mathcal{C}^{\eta}}\right\|_{L^{2p}(\Omega)}. X − X M , Λ ∞ L p ( Ω ) ≤ ∥ K M ∥ L 2 p ( Ω ) ζ f , b − ζ M , Λ C η L 2 p ( Ω ) . The first factor is bounded independently of M M M (68) with moment order 2 p 2p 2 p (74) .
□
Remark. Linear multiplicative diffusion also admits a direct quantitative bound. Use the balanced spectral assumptions of Theorem 4.3 : Λ = M 1 / 2 \Lambda=M^{1/2} Λ = M 1/2 δ λ ≤ C m e s h M − 1 / 2 \delta_\lambda\leq C_{\mathrm{mesh}}M^{-1/2} δ λ ≤ C mesh M − 1/2 ν , β ∈ R \nu,\beta\in\mathbb{R} ν , β ∈ R a ( x , t ) = ν x a(x,t)=\nu x a ( x , t ) = ν x σ ( x , t ) = β x \sigma(x,t)=\beta x σ ( x , t ) = β x X t = x 0 e ν t + β ζ t , f , b X_t=x_0e^{\nu t+\beta\zeta_{t,f,b}} X t = x 0 e ν t + β ζ t , f , b X t M , Λ = x 0 e ν t + β ζ t M , Λ X_t^{M,\Lambda}=x_0e^{\nu t+\beta\zeta_t^{M,\Lambda}} X t M , Λ = x 0 e ν t + β ζ t M , Λ
∥ X − X M , Λ ∥ ∞ ≤ ∣ x 0 β ∣ e ∣ ν ∣ T + ∣ β ∣ max { ∥ ζ f , b ∥ ∞ , ∥ ζ M , Λ ∥ ∞ } ∥ ζ f , b − ζ M , Λ ∥ ∞ . \|X-X^{M,\Lambda}\|_{\infty}\leq|x_0\beta|e^{|\nu|T+|\beta|\max\{\|\zeta_{f,b}\|_{\infty},\|\zeta^{M,\Lambda}\|_{\infty}\}}\|\zeta_{f,b}-\zeta^{M,\Lambda}\|_{\infty}. ∥ X − X M , Λ ∥ ∞ ≤ ∣ x 0 β ∣ e ∣ ν ∣ T + ∣ β ∣ m a x { ∥ ζ f , b ∥ ∞ , ∥ ζ M , Λ ∥ ∞ } ∥ ζ f , b − ζ M , Λ ∥ ∞ . Uniform Gaussian exponential bounds and Hölder’s inequality therefore yield, for p ≥ 2 p\geq2 p ≥ 2
( E ∥ X − X M , Λ ∥ ∞ p ) 1 / p ≤ C M − b / 4 p log ( 2 + M ) . \left(\mathbb{E}\|X-X^{M,\Lambda}\|_{\infty}^{p}\right)^{1/p}\leq CM^{-b/4p}\sqrt{\log(2+M)}. ( E ∥ X − X M , Λ ∥ ∞ p ) 1/ p ≤ C M − b /4 p log ( 2 + M ) . This uses the explicit linear solution, rather than the bounded-diffusion stability estimate of Lemma 4.5.
Remark 4.7. Our bounds use the exact cell masses and the maximum spectral cell width, allowing nonuniform partitions. For other quadrature rules, see the geometric partitions in [7 p. 99] and [13 Section 4] , Gaussian quadrature for the fractional kernel in [4 Section 2.1] , and the multifactor approximations of completely monotone kernels in [3 Sections 3–4] .
The preceding solution rate uses the joint Lipschitz regularity of ∂ x σ \partial_x\sigma ∂ x σ p p p L r L^r L r
Proposition 4.8 Let 0 < b < 2 0<b<2 0 < b < 2 T > 0 T>0 T > 0 f ∈ L ∞ ( 0 , T ) f\in L^\infty(0,T) f ∈ L ∞ ( 0 , T )
Λ M = { M 1 / 2 , 0 < b ≤ 1 , M 2 / 3 , 1 < b < 2. \Lambda_M=
\begin{cases}
M^{1/2}, & 0<b\leq1,\\
M^{2/3}, & 1<b<2.
\end{cases} Λ M = { M 1/2 , M 2/3 , 0 < b ≤ 1 , 1 < b < 2. Use spectral partitions whose maximum cell width satisfies δ λ , M ≤ C m e s h Λ M / M \delta_{\lambda,M} \le C_{\mathrm{mesh}}\Lambda_M/M δ λ , M ≤ C mesh Λ M / M C m e s h ≥ 1 C_{\mathrm{mesh}} \ge1 C mesh ≥ 1 x 0 x_0 x 0 1 ≤ r < ∞ 1 \le r < \infty 1 ≤ r < ∞
lim M → ∞ E [ sup 0 ≤ t ≤ T ∣ X t M , Λ M − X t ∣ r ] = 0. (76) \lim_{M\to\infty}\mathbb{E}\left[\sup_{0\le t\le T}\left|X_t^{M,\Lambda_M}-X_t\right|^r\right]=0. \tag*{(76)} M → ∞ lim E [ 0 ≤ t ≤ T sup X t M , Λ M − X t r ] = 0. (76) In particular, convergence is uniform in time in probability. For 1 < b < 2 1 < b < 2 1 < b < 2
Proof. First suppose 0 < b ≤ 1 0 < b \le1 0 < b ≤ 1
( H b ∘ ) − 1 < p < min ( 2 , 1 + α ) , p − 1 < η < η ′ < H b ∘ , (H_b^\circ)^{-1} < p < \min(2,1+\alpha), \qquad p^{-1}<\eta<\eta'<H_b^\circ, ( H b ∘ ) − 1 < p < min ( 2 , 1 + α ) , p − 1 < η < η ′ < H b ∘ , where H b ∘ H_b^\circ H b ∘ α \alpha α C η ( [ 0 , T ] ) C^\eta([0,T]) C η ([ 0 , T ]) C p -var ( [ 0 , T ] ) C^{p\text{-var}}([0,T]) C p -var ([ 0 , T ])
∥ z ∥ p -var ; [ 0 , T ] ≤ T η [ z ] η ; [ 0 , T ] , p η > 1. \|z\|_{p\text{-var};[0,T]} \le T^\eta[z]_{\eta;[0,T]}, \qquad p\eta>1. ∥ z ∥ p -var ; [ 0 , T ] ≤ T η [ z ] η ; [ 0 , T ] , p η > 1. Since the drivers start at zero, (68), used at moment order 2, implies convergence in probability in the p p p ∥ X M , Λ M − X ∥ ∞ → 0 \|X^{M,\Lambda_M}-X\|_\infty\to0 ∥ X M , Λ M − X ∥ ∞ → 0
For the moment estimate, let z z z η \eta η z 0 = 0 z_0=0 z 0 = 0 R = 1 + [ z ] η ; [ 0 , T ] R=1+[z]_{\eta;[0,T]} R = 1 + [ z ] η ; [ 0 , T ] x x x I = [ u , v ] I=[u,v] I = [ u , v ] h ≤ 1 h\le1 h ≤ 1 [9 Lemma 2.4, Eq. (2.8)] , applied with both variation exponents equal to p < 2 p<2 p < 2 σ \sigma σ
∥ x ∥ p -var ; I ≤ C ( h + R h η ) ( 1 + ∣ x u ∣ + ∥ x ∥ p -var ; I ) . \|x\|_{p\text{-var};I}\le C(h+Rh^\eta)(1+|x_u|+\|x\|_{p\text{-var};I}). ∥ x ∥ p -var ; I ≤ C ( h + R h η ) ( 1 + ∣ x u ∣ + ∥ x ∥ p -var ; I ) . Here both coefficient values grow at most linearly, and ∥ σ ( x ⋅ , ⋅ ) ∥ p -var ; I ≤ C ∥ x ∥ p -var ; I + C h \|\sigma(x_\cdot,\cdot)\|_{p\text{-var};I}\le C\|x\|_{p\text{-var};I}+Ch ∥ σ ( x ⋅ , ⋅ ) ∥ p -var ; I ≤ C ∥ x ∥ p -var ; I + C h h ≤ min { 1 , c R − 1 / η } h\le\min\{1,cR^{-1/\eta}\} h ≤ min { 1 , c R − 1/ η } c > 0 c>0 c > 0
∥ x ∥ p -var ; I ≤ C ( h + R h η ) ( 1 + ∣ x u ∣ ) . \|x\|_{p\text{-var};I}\le C(h+Rh^\eta)(1+|x_u|). ∥ x ∥ p -var ; I ≤ C ( h + R h η ) ( 1 + ∣ x u ∣ ) . Take an equal partition with mesh comparable to min { T , 1 , c R − 1 / η } \min\{T,1,cR^{-1/\eta}\} min { T , 1 , c R − 1/ η }
1 + ∣ x t j + 1 ∣ ≤ ( 1 + C [ h + R h η ] ) ( 1 + ∣ x t j ∣ ) . 1+|x_{t_{j+1}}|\le(1+C[h+Rh^\eta])(1+|x_{t_j}|). 1 + ∣ x t j + 1 ∣ ≤ ( 1 + C [ h + R h η ]) ( 1 + ∣ x t j ∣ ) . Since ∏ j ( 1 + C [ h + R h η ] ) ≤ exp ( C ∑ j [ h + R h η ] ) \prod_j(1+C[h+Rh^\eta])\le\exp(C\sum_j[h+Rh^\eta]) ∏ j ( 1 + C [ h + R h η ]) ≤ exp ( C ∑ j [ h + R h η ]) ∑ j R h η ≤ C T ( 1 + R 1 / η ) \sum_j Rh^\eta\le C_T(1+R^{1/\eta}) ∑ j R h η ≤ C T ( 1 + R 1/ η )
∥ x ∥ ∞ ; [ 0 , T ] ≤ C T ( 1 + ∣ x 0 ∣ ) exp ( C T [ 1 + [ z ] η ; [ 0 , T ] 1 / η ] ) . (77) \|x\|_{\infty;[0,T]}\le C_T(1+|x_0|)\exp\left(C_T[1+[z]_{\eta;[0,T]}^{1/\eta}]\right). \tag*{(77)} ∥ x ∥ ∞ ; [ 0 , T ] ≤ C T ( 1 + ∣ x 0 ∣ ) exp ( C T [ 1 + [ z ] η ; [ 0 , T ] 1/ η ] ) . (77) Since 1 / η < 2 1/\eta<2 1/ η < 2
sup M ≥ 1 E ∥ X M , Λ M ∥ ∞ q + E ∥ X ∥ ∞ q < ∞ for every 1 ≤ q < ∞ . \sup_{M\ge1}\mathbb{E}\|X^{M,\Lambda_M}\|_\infty^q+\mathbb{E}\|X\|_\infty^q<\infty\qquad\text{for every }1\le q<\infty. M ≥ 1 sup E ∥ X M , Λ M ∥ ∞ q + E ∥ X ∥ ∞ q < ∞ for every 1 ≤ q < ∞. Taking q > r q>r q > r ∥ X M , Λ M − X ∥ ∞ r \|X^{M,\Lambda_M}-X\|_\infty^r ∥ X M , Λ M − X ∥ ∞ r
For 1 < b < 2 1<b<2 1 < b < 2 r ≥ 2 r\ge2 r ≥ 2 1 ≤ r < 2 1\le r<2 1 ≤ r < 2 r = 2 r=2 r = 2
Remark 4.9 For fixed M M M Λ \Lambda Λ ( X M , Λ , Y M , Λ ) (X^{M,\Lambda},Y^{M,\Lambda}) ( X M , Λ , Y M , Λ ) C b C_b C b s s s ( x , y ) ∈ R × R M (x,y)\in\mathbb{R}\times\mathbb{R}^M ( x , y ) ∈ R × R M t ≥ s t\ge s t ≥ s
Y t k = e − ξ k ( t − s ) y k + ∫ s t e − ξ k ( t − u ) f ( u ) d B u k . Y_t^k=e^{-\xi_k(t-s)}y_k+\int_s^t e^{-\xi_k(t-u)}\sqrt{f(u)}\,\mathrm{d}B_u^k. Y t k = e − ξ k ( t − s ) y k + ∫ s t e − ξ k ( t − u ) f ( u ) d B u k . and the scalar component solves the ordinary integral equation with continuous velocity v t : = b ∑ k w k Y t k v_t:=\sqrt{b}\sum_k\sqrt{w_k}Y_t^k v t := b ∑ k w k Y t k F s \mathcal{F}_s F s
For the Feller property, couple two initial states using the same increments. The factor differences are deterministic:
Y t k − Y ~ t k = e − ξ k ( t − s ) ( y k − y ~ k ) . Y_t^k-\widetilde{Y}_t^k=e^{-\xi_k(t-s)}(y_k-\widetilde{y}_k). Y t k − Y t k = e − ξ k ( t − s ) ( y k − y k ) . On [ s , T ] [s,T] [ s , T ]
∥ X − X ~ ∥ ∞ ; [ s , T ] ≤ exp ( L a , T ( T − s ) + L σ , T ∥ v ∥ L 1 ( s , T ) ) ( ∣ x − x ~ ∣ + S X ~ , T ∥ v − v ~ ∥ L 1 ( s , T ) ) . \lVert X-\widetilde{X}\rVert_{\infty;[s,T]}
\le
\exp\left(L_{a,T}(T-s)+L_{\sigma,T}\lVert v\rVert_{L^1(s,T)}\right)
\left(|x-\widetilde{x}|+S_{\widetilde{X},T}\lVert v-\widetilde{v}\rVert_{L^1(s,T)}\right). ∥ X − X ∥ ∞ ; [ s , T ] ≤ exp ( L a , T ( T − s ) + L σ , T ∥ v ∥ L 1 ( s , T ) ) ( ∣ x − x ∣ + S X , T ∥ v − v ∥ L 1 ( s , T ) ) . Here L a , T L_{a,T} L a , T L σ , T L_{\sigma,T} L σ , T R × [ 0 , T ] \mathbb{R}\times[0,T] R × [ 0 , T ] S X ~ , T : = sup t ∈ [ s , T ] ∣ σ ( X ~ t , t ) ∣ S_{\widetilde{X},T}:=\sup_{t\in[s,T]}|\sigma(\widetilde{X}_t,t)| S X , T := sup t ∈ [ s , T ] ∣ σ ( X t , t ) ∣ S X ~ , T S_{\widetilde{X},T} S X , T v v v [ s , T ] [s,T] [ s , T ] ( x , y ) (x,y) ( x , y ) C b ( R × R M ) C_b(\mathbb{R}\times\mathbb{R}^M) C b ( R × R M )
Fully discrete approximation and the joint limit
We now discretize the finite-factor dynamics in time. The error has three contributions: the spectral approximation, Euler discretization of the Ornstein–Uhlenbeck factors, and left-point evaluation of the coefficients in the scalar equation. We keep the dependence on Λ \Lambda Λ
Fix M ∈ N M\in\mathbb{N} M ∈ N Λ > 0 \Lambda>0 Λ > 0 π = { 0 = t 0 < t 1 < ⋯ < t N = T } \pi=\{0=t_0<t_1<\cdots<t_N=T\} π = { 0 = t 0 < t 1 < ⋯ < t N = T } N ∈ N N\in\mathbb{N} N ∈ N
Δ n : = t n + 1 − t n , ∣ π ∣ : = max 0 ≤ n < N Δ n . \Delta_n:=t_{n+1}-t_n,\qquad|\pi|:=\max_{0\le n<N}\Delta_n. Δ n := t n + 1 − t n , ∣ π ∣ := 0 ≤ n < N max Δ n . and define the left-endpoint map η π ( t ) : = t n \eta_\pi(t):=t_n η π ( t ) := t n t ∈ [ t n , t n + 1 ) t\in[t_n,t_{n+1}) t ∈ [ t n , t n + 1 ) η π ( T ) : = t N − 1 \eta_\pi(T):=t_{N-1} η π ( T ) := t N − 1 T T T k = 1 , … , M k=1,\ldots,M k = 1 , … , M
W f k ( t ) : = ∫ 0 t f ( s ) d B s k . W_f^k(t):=\int_0^t\sqrt{f(s)}\,\mathrm{d}B_s^k. W f k ( t ) := ∫ 0 t f ( s ) d B s k . Define
Δ W f , n k : = W f k ( t n + 1 ) − W f k ( t n ) , 1 ≤ k ≤ M , 0 ≤ n < N . \Delta W_{f,n}^k:=W_f^k(t_{n+1})-W_f^k(t_n),\qquad1\le k\le M,\quad0\le n<N. Δ W f , n k := W f k ( t n + 1 ) − W f k ( t n ) , 1 ≤ k ≤ M , 0 ≤ n < N . The variables Δ W f , n k \Delta W^{k}_{f,n} Δ W f , n k
v f , n : = ∫ t n t n + 1 f ( s ) d s . v_{f,n} := \int_{t_n}^{t_{n+1}} f(s)\,\mathrm{d}s. v f , n := ∫ t n t n + 1 f ( s ) d s . Their simulation requires only the cell integrals v f , n v_{f,n} v f , n f f f
The continuous-time Euler approximation of the factors is
Y t k , M , Λ , π = − ξ k ∫ 0 t Y η π ( s ) k , M , Λ , π d s + W f k ( t ) , Y 0 k , M , Λ , π = 0. Y_t^{k,M,\Lambda,\pi}=-\xi_k\int_0^t Y_{\eta_\pi(s)}^{k,M,\Lambda,\pi}\,\mathrm{d}s+W_f^k(t),\qquad Y_0^{k,M,\Lambda,\pi}=0. Y t k , M , Λ , π = − ξ k ∫ 0 t Y η π ( s ) k , M , Λ , π d s + W f k ( t ) , Y 0 k , M , Λ , π = 0. At the grid points,
Y t n + 1 k , M , Λ , π = ( 1 − ξ k Δ n ) Y t n k , M , Λ , π + Δ W f , n k . (78) Y_{t_{n+1}}^{k,M,\Lambda,\pi}=(1-\xi_k\Delta_n)Y_{t_n}^{k,M,\Lambda,\pi}+\Delta W_{f,n}^k. \tag*{(78)} Y t n + 1 k , M , Λ , π = ( 1 − ξ k Δ n ) Y t n k , M , Λ , π + Δ W f , n k . (78) The aggregate factor process and its piecewise linear driver are
Y t M , Λ , π : = b ∑ k = 1 M w k Y t k , M , Λ , π \mathcal{Y}_t^{M,\Lambda,\pi}:=\sqrt{b}\sum_{k=1}^{M}\sqrt{w_k}Y_t^{k,M,\Lambda,\pi} Y t M , Λ , π := b k = 1 ∑ M w k Y t k , M , Λ , π and
ζ t M , Λ , π : = ∫ 0 t Y η π ( s ) M , Λ , π d s . \zeta_t^{M,\Lambda,\pi}:=\int_0^t\mathcal{Y}_{\eta_\pi(s)}^{M,\Lambda,\pi}\,\mathrm{d}s. ζ t M , Λ , π := ∫ 0 t Y η π ( s ) M , Λ , π d s . The Euler approximation of the scalar equation is
X t M , Λ , π = x 0 + ∫ 0 t a ( X η π ( s ) M , Λ , π , η π ( s ) ) d s + ∫ 0 t σ ( X η π ( s ) M , Λ , π , η π ( s ) ) Y η π ( s ) M , Λ , π d s . (79) X_t^{M,\Lambda,\pi}=x_0+\int_0^t a\left(X_{\eta_\pi(s)}^{M,\Lambda,\pi},\eta_\pi(s)\right)\,\mathrm{d}s+\int_0^t \sigma\left(X_{\eta_\pi(s)}^{M,\Lambda,\pi},\eta_\pi(s)\right)\mathcal{Y}_{\eta_\pi(s)}^{M,\Lambda,\pi}\,\mathrm{d}s. \tag*{(79)} X t M , Λ , π = x 0 + ∫ 0 t a ( X η π ( s ) M , Λ , π , η π ( s ) ) d s + ∫ 0 t σ ( X η π ( s ) M , Λ , π , η π ( s ) ) Y η π ( s ) M , Λ , π d s . (79) Equivalently,
X t n + 1 M , Λ , π = X t n M , Λ , π + a ( X t n M , Λ , π , t n ) Δ n + σ ( X t n M , Λ , π , t n ) Y t n M , Λ , π Δ n . X_{t_{n+1}}^{M,\Lambda,\pi}=X_{t_n}^{M,\Lambda,\pi}+a\left(X_{t_n}^{M,\Lambda,\pi},t_n\right)\Delta_n
+\sigma\left(X_{t_n}^{M,\Lambda,\pi},t_n\right)\mathcal{Y}_{t_n}^{M,\Lambda,\pi}\Delta_n. X t n + 1 M , Λ , π = X t n M , Λ , π + a ( X t n M , Λ , π , t n ) Δ n + σ ( X t n M , Λ , π , t n ) Y t n M , Λ , π Δ n . The restriction Λ ∣ π ∣ ≤ 1 \Lambda|\pi|\le1 Λ∣ π ∣ ≤ 1 0 ≤ 1 − ξ k Δ n ≤ 1 0\le1-\xi_k\Delta_n\le1 0 ≤ 1 − ξ k Δ n ≤ 1 Λ \Lambda Λ M M M
Proposition 4.10 Let T > 0 T>0 T > 0 0 < b < 2 0<b<2 0 < b < 2 f ∈ L ∞ ( 0 , T ) f\in L^\infty(0,T) f ∈ L ∞ ( 0 , T ) M ∈ N M\in\mathbb{N} M ∈ N Λ > 0 \Lambda>0 Λ > 0 p ≥ 2 p\ge2 p ≥ 2
Λ ∣ π ∣ ≤ 1. \Lambda|\pi|\le1. Λ∣ π ∣ ≤ 1. There is a constant C = C ( T , b , p , ∥ f ∥ ∞ , T ) C=C(T,b,p,\|f\|_{\infty,T}) C = C ( T , b , p , ∥ f ∥ ∞ , T ) M M M Λ \Lambda Λ π \pi π
max 1 ≤ k ≤ M sup t ≤ T ∥ Y t k , M , Λ − Y η π ( t ) k , M , Λ , π ∥ L p ( Ω ) ≤ C ∣ π ∣ 1 / 2 . (80) \max_{1\le k\le M}\sup_{t\le T}\left\|Y_t^{k,M,\Lambda}-Y_{\eta_\pi(t)}^{k,M,\Lambda,\pi}\right\|_{L^p(\Omega)}\le C|\pi|^{1/2}. \tag*{(80)} 1 ≤ k ≤ M max t ≤ T sup Y t k , M , Λ − Y η π ( t ) k , M , Λ , π L p ( Ω ) ≤ C ∣ π ∣ 1/2 . (80) and
sup t ≤ T ∥ Y t M , Λ − Y η π ( t ) M , Λ , π ∥ L p ( Ω ) ≤ C Λ ( 2 − b ) / 2 ∣ π ∣ 1 / 2 . (81) \sup_{t\le T}\left\|Y_t^{M,\Lambda}-Y_{\eta_\pi(t)}^{M,\Lambda,\pi}\right\|_{L^p(\Omega)}\le C\Lambda^{(2-b)/2}|\pi|^{1/2}. \tag*{(81)} t ≤ T sup Y t M , Λ − Y η π ( t ) M , Λ , π L p ( Ω ) ≤ C Λ ( 2 − b ) /2 ∣ π ∣ 1/2 . (81) Moreover, for every ρ ∈ ( 0 , 1 ) \rho\in(0,1) ρ ∈ ( 0 , 1 ) C ρ C_\rho C ρ
( E ∥ ζ M , Λ − ζ M , Λ , π ∥ C ( [ 0 , T ] ) p ) 1 / p ≤ C ρ Λ ( 2 − b ) / 2 ∣ π ∣ 1 / 2 . (82) \left(\mathbb{E}\left\|\zeta^{M,\Lambda}-\zeta^{M,\Lambda,\pi}\right\|_{C([0,T])}^{p}\right)^{1/p}\le C_\rho\Lambda^{(2-b)/2}|\pi|^{1/2}. \tag*{(82)} ( E ζ M , Λ − ζ M , Λ , π C ([ 0 , T ]) p ) 1/ p ≤ C ρ Λ ( 2 − b ) /2 ∣ π ∣ 1/2 . (82) Proof. The first two bounds place the supremum outside the probability norm. We obtain the path-norm bound by integrating the aggregate error in time.
Write h : = ∣ π ∣ h:=|\pi| h := ∣ π ∣ ξ = ξ k \xi=\xi_k ξ = ξ k
Y t n k , M , Λ , π = ∑ j = 0 n − 1 ( ∏ ℓ = j + 1 n − 1 ( 1 − ξ Δ ℓ ) ) Δ W f , j k . Y_{t_n}^{k,M,\Lambda,\pi}=\sum_{j=0}^{n-1}\left(\prod_{\ell=j+1}^{n-1}(1-\xi\Delta_\ell)\right)\Delta W_{f,j}^{k}. Y t n k , M , Λ , π = j = 0 ∑ n − 1 ℓ = j + 1 ∏ n − 1 ( 1 − ξ Δ ℓ ) Δ W f , j k . Since 0 ≤ 1 − ξ Δ ℓ ≤ 1 0\le1-\xi\Delta_\ell\le1 0 ≤ 1 − ξ Δ ℓ ≤ 1 e − ξ ( t n − t j + 1 ) e^{-\xi(t_n-t_{j+1})} e − ξ ( t n − t j + 1 )
E ∣ Y t n k , M , Λ , π ∣ 2 ≤ ∥ f ∥ ∞ , T 2 ∑ j = 0 n − 1 e − 2 ξ ( t n − t j + 1 ) Δ j ≤ ∥ f ∥ ∞ , T 2 min { T , e 2 2 ξ } . \begin{aligned}
\mathbb{E}\left|Y_{t_n}^{k,M,\Lambda,\pi}\right|^2\le\|f\|_{\infty,T}^2\sum_{j=0}^{n-1}e^{-2\xi(t_n-t_{j+1})}\Delta_j \\
&\le\|f\|_{\infty,T}^2\min\left\{T,\frac{e^2}{2\xi}\right\}.
\end{aligned} E Y t n k , M , Λ , π 2 ≤ ∥ f ∥ ∞ , T 2 j = 0 ∑ n − 1 e − 2 ξ ( t n − t j + 1 ) Δ j ≤ ∥ f ∥ ∞ , T 2 min { T , 2 ξ e 2 } . Indeed, comparison on each mesh cell and ξ h ≤ 1 \xi h\le1 ξ h ≤ 1
∑ j = 0 n − 1 e − 2 ξ ( t n − t j + 1 ) Δ j ≤ e 2 ξ h ∫ 0 t n e − 2 ξ ( t n − s ) d s ≤ e 2 2 ξ ; \sum_{j=0}^{n-1}e^{-2\xi(t_n-t_{j+1})}\Delta_j\le e^{2\xi h}\int_0^{t_n}e^{-2\xi(t_n-s)}\,\mathrm{d}s\le\frac{e^2}{2\xi}; j = 0 ∑ n − 1 e − 2 ξ ( t n − t j + 1 ) Δ j ≤ e 2 ξ h ∫ 0 t n e − 2 ξ ( t n − s ) d s ≤ 2 ξ e 2 ; the alternative bound by T T T
For 0 ≤ t ≤ T 0\le t\le T 0 ≤ t ≤ T
R t k : = Y t k , M , Λ , π − Y η π ( t ) k , M , Λ , π . R_t^k:=Y_t^{k,M,\Lambda,\pi}-Y_{\eta_\pi(t)}^{k,M,\Lambda,\pi}. R t k := Y t k , M , Λ , π − Y η π ( t ) k , M , Λ , π . On a cell with left endpoint t n = η π ( t ) t_n=\eta_\pi(t) t n = η π ( t )
R t k = − ξ ( t − t n ) Y t n k , M , Λ , π + W f k ( t ) − W f k ( t n ) . R_t^k=-\xi(t-t_n)Y_{t_n}^{k,M,\Lambda,\pi}+W_f^k(t)-W_f^k(t_n). R t k = − ξ ( t − t n ) Y t n k , M , Λ , π + W f k ( t ) − W f k ( t n ) . The drift contribution has L p L^p L p C ξ h min { T 1 / 2 , ξ − 1 / 2 } ≤ C h ξ ≤ C h 1 / 2 C\xi h\min\{T^{1/2},\xi^{-1/2}\}\le Ch\sqrt{\xi}\le Ch^{1/2} C ξ h min { T 1/2 , ξ − 1/2 } ≤ C h ξ ≤ C h 1/2 ∥ f ∥ ∞ , T h \|f\|_{\infty,T}h ∥ f ∥ ∞ , T h
sup t ≤ T ∥ R t k ∥ L p ( Ω ) ≤ C h 1 / 2 , (83) \sup_{t\le T}\left\|R_t^k\right\|_{L^p(\Omega)}\le Ch^{1/2}, \tag*{(83)} t ≤ T sup R t k L p ( Ω ) ≤ C h 1/2 , (83) uniformly in ξ \xi ξ
Let E t k : = Y t k , M , Λ − Y t k , M , Λ , π E_t^k:=Y_t^{k,M,\Lambda}-Y_t^{k,M,\Lambda,\pi} E t k := Y t k , M , Λ − Y t k , M , Λ , π
E t k = − ξ ∫ 0 t e − ξ ( t − s ) R s k d s . E_t^k=-\xi\int_0^t e^{-\xi(t-s)}R_s^k\,\mathrm{d}s. E t k = − ξ ∫ 0 t e − ξ ( t − s ) R s k d s . Since ξ ∫ t 0 e − ξ ( t − s ) d s ≤ 1 \xi\int_t^0 e^{-\xi(t-s)}\,\mathrm{d}s \le1 ξ ∫ t 0 e − ξ ( t − s ) d s ≤ 1 sup t ≤ T ∥ E t k ∥ L p ( Ω ) ≤ C h 1 / 2 \sup_{t\le T}\|E_t^k\|_{L^p(\Omega)} \le Ch^{1/2} sup t ≤ T ∥ E t k ∥ L p ( Ω ) ≤ C h 1/2 Y t k , M , Λ − Y η π ( t ) k , M , Λ , π = E t k + R t k Y_t^{k,M,\Lambda}-Y_{\eta_\pi(t)}^{k,M,\Lambda,\pi}=E_t^k+R_t^k Y t k , M , Λ − Y η π ( t ) k , M , Λ , π = E t k + R t k
The errors for distinct k k k
∥ Y t M , Λ − Y η π ( t ) M , Λ , π ∥ L p ( Ω ) 2 ≤ C p h ∑ k = 1 M w k = C p h Λ 2 − b ( 2 − b ) Γ ( 2 − b ) . (84) \left\|Y_t^{M,\Lambda}-Y_{\eta_\pi(t)}^{M,\Lambda,\pi}\right\|_{L^p(\Omega)}^2 \le C_p h\sum_{k=1}^{M}w_k=C_p h\frac{\Lambda^{2-b}}{(2-b)\Gamma(2-b)}. \tag*{(84)} Y t M , Λ − Y η π ( t ) M , Λ , π L p ( Ω ) 2 ≤ C p h k = 1 ∑ M w k = C p h ( 2 − b ) Γ ( 2 − b ) Λ 2 − b . (84) which is (81).
Independence makes the variances add, so the aggregate error carries the square root of the total spectral mass ∑ k w k = μ b ( ( 0 , Λ ] ) \sum_k w_k=\mu_b((0,\Lambda]) ∑ k w k = μ b (( 0 , Λ ]) Λ ( 2 − b ) / 2 \Lambda^{(2-b)/2} Λ ( 2 − b ) /2 M M M
The driver error vanishes at zero and has almost-everywhere derivative
D t : = Y t M , Λ − Y η π ( t ) M , Λ , π . D_t:=Y_t^{M,\Lambda}-Y_{\eta_\pi(t)}^{M,\Lambda,\pi}. D t := Y t M , Λ − Y η π ( t ) M , Λ , π . Choose q 0 > 1 / ( 1 − ρ ) q_0>1/(1-\rho) q 0 > 1/ ( 1 − ρ ) r : = max { p , q 0 } r:=\max\{p,q_0\} r := max { p , q 0 }
∣ ∫ s t D u d u ∣ ≤ ∣ t − s ∣ 1 − 1 / q 0 ∥ D ∥ L q 0 ( 0 , T ) . \left|\int_s^t D_u\,\mathrm{d}u\right|\le|t-s|^{1-1/q_0}\|D\|_{L^{q_0}(0,T)}. ∫ s t D u d u ≤ ∣ t − s ∣ 1 − 1/ q 0 ∥ D ∥ L q 0 ( 0 , T ) . Since 1 − 1 / q 0 > ρ 1-1/q_0>\rho 1 − 1/ q 0 > ρ C ρ C^\rho C ρ C T , ρ , q 0 ∥ D ∥ L q 0 ( 0 , T ) C_{T,\rho,q_0}\|D\|_{L^{q_0}(0,T)} C T , ρ , q 0 ∥ D ∥ L q 0 ( 0 , T ) r ≥ p , q 0 r\ge p,q_0 r ≥ p , q 0
∥ D ∥ L p ( Ω ; L q 0 ( 0 , T ) ) ≤ T 1 / q 0 sup t ≤ T ∥ D t ∥ L r ( Ω ) . \|D\|_{L^p(\Omega;L^{q_0}(0,T))}\le T^{1/q_0}\sup_{t\le T}\|D_t\|_{L^r(\Omega)}. ∥ D ∥ L p ( Ω ; L q 0 ( 0 , T )) ≤ T 1/ q 0 t ≤ T sup ∥ D t ∥ L r ( Ω ) . Applying (81) at moment order r r r □ \square □
Assumption 4.11. Assume Assumption 4.4. For every T > 0 T>0 T > 0 K T , L T K_T,L_T K T , L T
∣ a ( x , t ) ∣ ≤ K T , |a(x,t)|\le K_T, ∣ a ( x , t ) ∣ ≤ K T , ∣ a ( x , t ) − a ( y , s ) ∣ ≤ L T ( ∣ x − y ∣ + ∣ t − s ∣ ) , x , y ∈ R , s , t ∈ [ 0 , T ] . |a(x,t)-a(y,s)|\le L_T(|x-y|+|t-s|),\qquad x,y\in\mathbb{R},\quad s,t\in[0,T]. ∣ a ( x , t ) − a ( y , s ) ∣ ≤ L T ( ∣ x − y ∣ + ∣ t − s ∣ ) , x , y ∈ R , s , t ∈ [ 0 , T ] . The bounded drift makes the flow estimates uniform in the initial value; its additional Lipschitz regularity in time controls the drift-freezing error. The next lemma bounds each local Euler defect by Δ n 2 η \Delta_n^{2\eta} Δ n 2 η ∣ π ∣ 2 η − 1 |\pi|^{2\eta-1} ∣ π ∣ 2 η − 1 1 / η 1/\eta 1/ η
Lemma 4.12. Let T > 0 T>0 T > 0 η ∈ ( 1 / 2 , 1 ) \eta\in(1/2,1) η ∈ ( 1/2 , 1 ) z ∈ C η ( [ 0 , T ] ) z\in C^\eta([0,T]) z ∈ C η ([ 0 , T ]) z ( 0 ) = 0 z(0)=0 z ( 0 ) = 0 x x x
x t = x 0 + ∫ 0 t a ( x s , s ) d s + ∫ 0 t σ ( x s , s ) d z s . x_t=x_0+\int_0^t a(x_s,s)\,\mathrm{d}s+\int_0^t \sigma(x_s,s)\,\mathrm{d}z_s. x t = x 0 + ∫ 0 t a ( x s , s ) d s + ∫ 0 t σ ( x s , s ) d z s . For t ∈ [ t n , t n + 1 ] t\in[t_n,t_{n+1}] t ∈ [ t n , t n + 1 ]
x t π = x t n π + a ( x t n π , t n ) ( t − t n ) + σ ( x t n π , t n ) ( z t − z t n ) , x 0 π = x 0 . (85) x_t^\pi=x_{t_n}^\pi+a(x_{t_n}^\pi,t_n)(t-t_n)+\sigma(x_{t_n}^\pi,t_n)(z_t-z_{t_n}),\qquad x_0^\pi=x_0. \tag*{(85)} x t π = x t n π + a ( x t n π , t n ) ( t − t n ) + σ ( x t n π , t n ) ( z t − z t n ) , x 0 π = x 0 . (85) Then
∥ x − x π ∥ ∞ ; [ 0 , T ] ≤ C exp ( C [ 1 + ∥ z ∥ C η ( [ 0 , T ] ) 1 / η ] ) ∣ π ∣ 2 η − 1 . (86) \|x-x^\pi\|_{\infty;[0,T]}\le C\exp\left(C\left[1+\|z\|_{C^\eta([0,T])}^{1/\eta}\right]\right)|\pi|^{2\eta-1}. \tag*{(86)} ∥ x − x π ∥ ∞ ; [ 0 , T ] ≤ C exp ( C [ 1 + ∥ z ∥ C η ([ 0 , T ]) 1/ η ] ) ∣ π ∣ 2 η − 1 . (86) The constant C C C T T T η \eta η R × [ 0 , T ] \mathbb{R}\times[0,T] R × [ 0 , T ]
Proof. Put K : = 1 + [ z ] η ; [ 0 , T ] K := 1 + [z]_{\eta;[0,T]} K := 1 + [ z ] η ; [ 0 , T ] T T T η \eta η φ s , t ( ξ ) \varphi_{s,t}(\xi) φ s , t ( ξ ) t t t ξ \xi ξ s s s z z z
sup 0 ≤ s ≤ t ≤ T Lip ( φ s , t ) ≤ C exp ( C K 1 / η ) . (87) \sup_{0 \le s \le t \le T} \operatorname{Lip}(\varphi_{s,t}) \le C\exp(CK^{1/\eta}). \tag*{(87)} 0 ≤ s ≤ t ≤ T sup Lip ( φ s , t ) ≤ C exp ( C K 1/ η ) . (87) The time shifts in this application preserve the coefficient bounds, and the shifted driver has Hölder norm at most C T K C_TK C T K
We first estimate the error of a single Euler step uniformly in its initial value. The bounded-drift case of the local solution estimate (72) gives
[ φ s , ⋅ ( ξ ) ] η ; [ s , s + ℓ ] ≤ C K if ℓ ≤ d : = min { T , c K − 1 / η } , [\varphi_{s,\cdot}(\xi)]_{\eta;[s,s+\ell]} \le CK \quad\text{if } \ell\le d := \min\{T,cK^{-1/\eta}\}, [ φ s , ⋅ ( ξ ) ] η ; [ s , s + ℓ ] ≤ C K if ℓ ≤ d := min { T , c K − 1/ η } , where 0 < c ≤ 1 0 < c \le1 0 < c ≤ 1 C K ℓ η ≤ C CK\ell^\eta\le C C K ℓ η ≤ C C T ( 1 + K 1 / η ) C_T(1+K^{1/\eta}) C T ( 1 + K 1/ η )
sup s ≤ t , ξ ∣ φ s , t ( ξ ) − ξ ∣ ≤ C ( 1 + K 1 / η ) . (88) \sup_{s \le t,\xi} |\varphi_{s,t}(\xi)-\xi| \le C(1+K^{1/\eta}). \tag*{(88)} s ≤ t , ξ sup ∣ φ s , t ( ξ ) − ξ ∣ ≤ C ( 1 + K 1/ η ) . (88) Write
Ψ s , t ( ξ ) : = ξ + a ( ξ , s ) ( t − s ) + σ ( ξ , s ) ( z t − z s ) , Δ : = t − s . \Psi_{s,t}(\xi):=\xi+a(\xi,s)(t-s)+\sigma(\xi,s)(z_t-z_s), \qquad\Delta:=t-s. Ψ s , t ( ξ ) := ξ + a ( ξ , s ) ( t − s ) + σ ( ξ , s ) ( z t − z s ) , Δ := t − s . For Δ ≤ d \Delta\le d Δ ≤ d x r = φ s , r ( ξ ) x_r=\varphi_{s,r}(\xi) x r = φ s , r ( ξ )
φ s , t ( ξ ) − Ψ s , t ( ξ ) = ∫ s t [ a ( x r , r ) − a ( ξ , s ) ] d r + ∫ s t [ σ ( x r , r ) − σ ( ξ , s ) ] d z r . \varphi_{s,t}(\xi)-\Psi_{s,t}(\xi)
=\int_s^t [a(x_r,r)-a(\xi,s)]\,\mathrm{d}r
+\int_s^t [\sigma(x_r,r)-\sigma(\xi,s)]\,\mathrm{d}z_r. φ s , t ( ξ ) − Ψ s , t ( ξ ) = ∫ s t [ a ( x r , r ) − a ( ξ , s )] d r + ∫ s t [ σ ( x r , r ) − σ ( ξ , s )] d z r . Joint Lipschitz continuity of the drift bounds the first term by C ( K Δ 1 + η + Δ 2 ) C(K\Delta^{1+\eta}+\Delta^2) C ( K Δ 1 + η + Δ 2 ) s s s η \eta η C K CK C K C K 2 Δ 2 η CK^2\Delta^{2\eta} C K 2 Δ 2 η η < 1 \eta<1 η < 1 Δ ≤ T \Delta\le T Δ ≤ T C T K 2 Δ 2 η C_TK^2\Delta^{2\eta} C T K 2 Δ 2 η Δ ≤ d \Delta\le d Δ ≤ d
If Δ > d \Delta>d Δ > d d = c K − 1 / η d=cK^{-1/\eta} d = c K − 1/ η (88) give a defect at most C T ( 1 + K 1 / η ) C_T(1+K^{1/\eta}) C T ( 1 + K 1/ η ) Δ 2 η > c 2 η K − 2 \Delta^{2\eta}>c^{2\eta}K^{-2} Δ 2 η > c 2 η K − 2 C T K 2 + 1 / η Δ 2 η C_TK^{2+1/\eta}\Delta^{2\eta} C T K 2 + 1/ η Δ 2 η K ≥ 1 K\ge1 K ≥ 1 C exp ( C K 1 / η ) Δ 2 η C\exp(CK^{1/\eta})\Delta^{2\eta} C exp ( C K 1/ η ) Δ 2 η
sup ξ ∣ φ s , t ( ξ ) − Ψ s , t ( ξ ) ∣ ≤ C exp ( C K 1 / η ) Δ 2 η , 0 ≤ s < t ≤ T . (89) \sup_{\xi}|\varphi_{s,t}(\xi)-\Psi_{s,t}(\xi)|
\le C\exp(CK^{1/\eta})\Delta^{2\eta}, \qquad0\le s<t\le T. \tag*{(89)} ξ sup ∣ φ s , t ( ξ ) − Ψ s , t ( ξ ) ∣ ≤ C exp ( C K 1/ η ) Δ 2 η , 0 ≤ s < t ≤ T . (89) Thus no restriction on the temporal mesh is needed for this pathwise bound.
For a fixed mesh point t m t_m t m F k : = φ t k , t m ( x t k π ) F_k:=\varphi_{t_k,t_m}(x^\pi_{t_k}) F k := φ t k , t m ( x t k π ) k = 0 , … , m k=0,\ldots,m k = 0 , … , m F 0 = x t m F_0=x_{t_m} F 0 = x t m F m = x t m π F_m=x^\pi_{t_m} F m = x t m π
∣ F k − F k + 1 ∣ ≤ Lip ( φ t k + 1 , t m ) ∣ φ t k , t k + 1 ( x t k π ) − Ψ t k , t k + 1 ( x t k π ) ∣ . |F_k-F_{k+1}|
\le\operatorname{Lip}(\varphi_{t_{k+1},t_m})
\left|\varphi_{t_k,t_{k+1}}(x^\pi_{t_k})-\Psi_{t_k,t_{k+1}}(x^\pi_{t_k})\right|. ∣ F k − F k + 1 ∣ ≤ Lip ( φ t k + 1 , t m ) φ t k , t k + 1 ( x t k π ) − Ψ t k , t k + 1 ( x t k π ) . Summing these differences and using (87) and (89) gives
max k ∣ x t m − x t m π ∣ ≤ C exp ( C K 1 / η ) ∑ k Δ k 2 η ≤ C T exp ( C K 1 / η ) ∣ π ∣ 2 η − 1 . (90) \max_k |x_{t_m}-x^\pi_{t_m}|
\le C\exp(CK^{1/\eta})\sum_k\Delta_k^{2\eta}
\le C_T\exp(CK^{1/\eta})|\pi|^{2\eta-1}. \tag*{(90)} k max ∣ x t m − x t m π ∣ ≤ C exp ( C K 1/ η ) k ∑ Δ k 2 η ≤ C T exp ( C K 1/ η ) ∣ π ∣ 2 η − 1 . (90) This telescoping uses exact flows after each local error, so it does not require a Hölder estimate for the numerical path.
To include times between the nodes, the local estimate and (88) give [ x ] η ; [ 0 , T ] ≤ C ( 1 + K 1 + 1 / η ) [x]_{\eta;[0,T]}\le C(1+K^{1+1/\eta}) [ x ] η ; [ 0 , T ] ≤ C ( 1 + K 1 + 1/ η ) d d d d η d^\eta d η
∣ x t π − x t n π ∣ ≤ C T K ∣ π ∣ η , t n ≤ t ≤ t n + 1 . |x^\pi_t-x^\pi_{t_n}|\le C_TK|\pi|^\eta,\qquad t_n\le t\le t_{n+1}. ∣ x t π − x t n π ∣ ≤ C T K ∣ π ∣ η , t n ≤ t ≤ t n + 1 . Hence the uniform error is bounded by the nodal error plus C ( 1 + K 1 + 1 / η ) ∣ π ∣ η C(1+K^{1+1/\eta})|\pi|^\eta C ( 1 + K 1 + 1/ η ) ∣ π ∣ η η > 2 η − 1 \eta>2\eta-1 η > 2 η − 1 ∣ π ∣ ≤ T |\pi|\le T ∣ π ∣ ≤ T K K K
We combine the spectral and temporal estimates next. In the finite-variation regime we use Gronwall’s inequality; in the Young regime we use Lemmas 4.5 and 4.12. The three terms in (91) bound the spectral, factor discretization, and coefficient-freezing errors, respectively.
Theorem 4.13 Let T > 0 T>0 T > 0 p ≥ 2 p\ge2 p ≥ 2 f ∈ L ∞ ( 0 , T ) f\in L^\infty(0,T) f ∈ L ∞ ( 0 , T ) δ λ ≤ C m e s h Λ / M \delta_\lambda\le C_{\mathrm{mesh}}\Lambda/M δ λ ≤ C mesh Λ/ M C m e s h ≥ 1 C_{\mathrm{mesh}}\ge1 C mesh ≥ 1 ξ k ∈ I k \xi_k\in I_k ξ k ∈ I k W W W x 0 x_0 x 0 0 < b ≤ 1 0<b\le1 0 < b ≤ 1 1 < b < 2 1<b<2 1 < b < 2 R × [ 0 , T ] \mathbb{R}\times[0,T] R × [ 0 , T ]
For 1 < b < 2 1<b<2 1 < b < 2 Λ = M 2 / 3 \Lambda=M^{2/3} Λ = M 2/3 M 2 / 3 ∣ π ∣ ≤ 1 M^{2/3}|\pi|\le1 M 2/3 ∣ π ∣ ≤ 1
( E sup t ≤ T ∣ X t − X t M , Λ , π ∣ p ) 1 / p ≤ C M − ( b − 1 ) / 3 + C M ( 2 − b ) / 3 ∣ π ∣ 1 / 2 ≤ C M − ( b − 1 ) / 3 . (91) \left(\mathbb{E}\sup_{t\le T}\left|X_t-X_t^{M,\Lambda,\pi}\right|^p\right)^{1/p}
\le CM^{-(b-1)/3}+CM^{(2-b)/3}|\pi|^{1/2}\le CM^{-(b-1)/3}. \tag*{(91)} ( E t ≤ T sup X t − X t M , Λ , π p ) 1/ p ≤ C M − ( b − 1 ) /3 + C M ( 2 − b ) /3 ∣ π ∣ 1/2 ≤ C M − ( b − 1 ) /3 . (91) For 0 < b ≤ 1 0<b\le1 0 < b ≤ 1 Λ = M 1 / 2 \Lambda=M^{1/2} Λ = M 1/2
1 2 < η < η ′ < H b ∘ . \frac{1}{2}<\eta<\eta'<H_b^\circ. 2 1 < η < η ′ < H b ∘ . If M ( 2 − b ) / 2 ∣ π ∣ ≤ 1 M^{(2-b)/2}|\pi|\le1 M ( 2 − b ) /2 ∣ π ∣ ≤ 1
( E sup t ≤ T ∣ X t − X t M , Λ , π ∣ p ) 1 / p ≤ C ( M − b / 4 log ( 2 + M ) ) 1 − η / η ′ + C M ( 2 − b ) / 4 ∣ π ∣ 1 / 2 + C ∣ π ∣ 2 η − 1 . (92) \left(\mathbb{E}\sup_{t\le T}\left|X_t-X_t^{M,\Lambda,\pi}\right|^p\right)^{1/p}
\le C\left(M^{-b/4}\sqrt{\log(2+M)}\right)^{1-\eta/\eta'}+CM^{(2-b)/4}|\pi|^{1/2}+C|\pi|^{2\eta-1}. \tag*{(92)} ( E t ≤ T sup X t − X t M , Λ , π p ) 1/ p ≤ C ( M − b /4 log ( 2 + M ) ) 1 − η / η ′ + C M ( 2 − b ) /4 ∣ π ∣ 1/2 + C ∣ π ∣ 2 η − 1 . (92) All constants in (90)–(91) may depend on C m e s h C_{\mathrm{mesh}} C mesh M M M π \pi π
Proof In the finite-variation case, abbreviate
Z : = Y M , Λ , Z π : = Y η π ( ⋅ ) M , Λ , π . Z:=Y^{M,\Lambda},\qquad Z^\pi:=Y_{\eta^\pi(\cdot)}^{M,\Lambda,\pi}. Z := Y M , Λ , Z π := Y η π ( ⋅ ) M , Λ , π . Proposition 4.10 gives
∥ Z − Z π ∥ L p ( Ω ; L 1 ( 0 , T ) ) ≤ C M ( 2 − b ) / 3 ∣ π ∣ 1 / 2 . (93) \|Z-Z^\pi\|_{L^p(\Omega;L^1(0,T))}\le CM^{(2-b)/3}|\pi|^{1/2}. \tag*{(93)} ∥ Z − Z π ∥ L p ( Ω ; L 1 ( 0 , T )) ≤ C M ( 2 − b ) /3 ∣ π ∣ 1/2 . (93) Theorem 4.1 and the triangle inequality imply that the centered Gaussian random elements Z Z Z L 2 ( Ω ; L 2 ( 0 , T ) ) L^2(\Omega;L^2(0,T)) L 2 ( Ω ; L 2 ( 0 , T )) p = 2 p=2 p = 2 M 2 / 3 ∣ π ∣ ≤ 1 M^{2/3}|\pi|\le1 M 2/3 ∣ π ∣ ≤ 1 Z − Z π Z-Z^\pi Z − Z π Z π Z^\pi Z π U ∈ { Z , Z π } U\in\{Z,Z^\pi\} U ∈ { Z , Z π } Q M , π Q_{M,\pi} Q M , π L 2 ( 0 , T ) L^2(0,T) L 2 ( 0 , T )
∥ Q M , π ∥ o p ≤ Tr Q M , π = E ∥ U ∥ L 2 ( 0 , T ) 2 ≤ C T \|Q_{M,\pi}\|_{\mathrm{op}}\le\operatorname{Tr}Q_{M,\pi}=\mathbb{E}\|U\|_{L^2(0,T)}^2\le C_T ∥ Q M , π ∥ op ≤ Tr Q M , π = E ∥ U ∥ L 2 ( 0 , T ) 2 ≤ C T uniformly in M M M π \pi π 0 < ε F < ( 4 C T ) − 1 0 < \varepsilon_F < (4C_T)^{-1} 0 < ε F < ( 4 C T ) − 1 ( λ j ) (\lambda_j) ( λ j ) Q M , π Q_{M,\pi} Q M , π
E e ε F ∥ U ∥ L 2 ( 0 , T ) 2 = ∏ j ( 1 − 2 ε F λ j ) − 1 / 2 ≤ exp ( ε F Tr Q M , π 1 − 2 ε F ∥ Q M , π ∥ o p ) ≤ e 2 ε F C T . \mathbb{E}e^{\varepsilon_F\|U\|_{L^2(0,T)}^2}=\prod_j(1-2\varepsilon_F\lambda_j)^{-1/2}\leq\exp\left(\frac{\varepsilon_F\operatorname{Tr}Q_{M,\pi}}{1-2\varepsilon_F\|Q_{M,\pi}\|_{\mathrm{op}}}\right)\leq e^{2\varepsilon_F C_T}. E e ε F ∥ U ∥ L 2 ( 0 , T ) 2 = j ∏ ( 1 − 2 ε F λ j ) − 1/2 ≤ exp ( 1 − 2 ε F ∥ Q M , π ∥ op ε F Tr Q M , π ) ≤ e 2 ε F C T . The product formula follows from the orthonormal eigenbasis expansion and monotone convergence. Thus Z Z Z Z π Z^\pi Z π L 2 ( 0 , T ) L^2(0,T) L 2 ( 0 , T ) ∥ U ∥ L 1 ≤ T ∥ U ∥ L 2 \|U\|_{L^1}\leq\sqrt{T}\|U\|_{L^2} ∥ U ∥ L 1 ≤ T ∥ U ∥ L 2 L 1 ( 0 , T ) L^1(0,T) L 1 ( 0 , T )
Let
D t : = sup r ≤ t ∣ X r M , Λ − X r M , Λ , π ∣ . D_t:=\sup_{r\leq t}\left|X_r^{M,\Lambda}-X_r^{M,\Lambda,\pi}\right|. D t := r ≤ t sup X r M , Λ − X r M , Λ , π . On every mesh cell,
∣ X s M , Λ , π − X η π ( s ) M , Λ , π ∣ ≤ C ∣ π ∣ ( 1 + ∣ Z s π ∣ ) . \left|X_s^{M,\Lambda,\pi}-X_{\eta_\pi(s)}^{M,\Lambda,\pi}\right|\leq C|\pi|(1+|Z_s^\pi|). X s M , Λ , π − X η π ( s ) M , Λ , π ≤ C ∣ π ∣ ( 1 + ∣ Z s π ∣ ) . Subtracting the two scalar equations and using the joint Lipschitz bounds in space and time gives
D t ≤ C ∫ 0 t ( 1 + ∣ Z s ∣ ) D s d s D_t\leq C\int_0^t(1+|Z_s|)D_s\,\mathrm{d}s D t ≤ C ∫ 0 t ( 1 + ∣ Z s ∣ ) D s d s + C ∣ π ∣ ∫ 0 T ( 1 + ∣ Z s ∣ ) ( 1 + ∣ Z s π ∣ ) d s + C ∫ 0 T ∣ Z s − Z s π ∣ d s . {}+C|\pi|\int_0^T(1+|Z_s|)(1+|Z_s^\pi|)\,\mathrm{d}s+C\int_0^T|Z_s-Z_s^\pi|\,\mathrm{d}s. + C ∣ π ∣ ∫ 0 T ( 1 + ∣ Z s ∣ ) ( 1 + ∣ Z s π ∣ ) d s + C ∫ 0 T ∣ Z s − Z s π ∣ d s . The local error term satisfies
∫ 0 T ( 1 + ∣ Z s ∣ ) ( 1 + ∣ Z s π ∣ ) d s ≤ T + T ( ∥ Z ∥ L 2 + ∥ Z π ∥ L 2 ) + ∥ Z ∥ L 2 ∥ Z π ∥ L 2 , \int_0^T(1+|Z_s|)(1+|Z_s^\pi|)\,\mathrm{d}s\leq T+\sqrt{T}(\|Z\|_{L^2}+\|Z^\pi\|_{L^2})+\|Z\|_{L^2}\|Z^\pi\|_{L^2}, ∫ 0 T ( 1 + ∣ Z s ∣ ) ( 1 + ∣ Z s π ∣ ) d s ≤ T + T ( ∥ Z ∥ L 2 + ∥ Z π ∥ L 2 ) + ∥ Z ∥ L 2 ∥ Z π ∥ L 2 , whose right-hand side has uniformly bounded moments of every order. Gronwall’s inequality, the exponential moment bounds, Hölder’s inequality, and (93) at a higher moment order therefore give
( E sup t ≤ T ∣ X t M , Λ − X t M , Λ , π ∣ p ) 1 / p ≤ C M ( 2 − b ) / 3 ∣ π ∣ 1 / 2 . \left(\mathbb{E}\sup_{t\leq T}\left|X_t^{M,\Lambda}-X_t^{M,\Lambda,\pi}\right|^p\right)^{1/p}\leq CM^{(2-b)/3}|\pi|^{1/2}. ( E t ≤ T sup X t M , Λ − X t M , Λ , π p ) 1/ p ≤ C M ( 2 − b ) /3 ∣ π ∣ 1/2 . Here the term of order ∣ π ∣ |\pi| ∣ π ∣ ∣ π ∣ ≤ ∣ π ∣ 1 / 2 ≤ M ( 2 − b ) / 3 ∣ π ∣ 1 / 2 |\pi|\leq|\pi|^{1/2}\leq M^{(2-b)/3}|\pi|^{1/2} ∣ π ∣ ≤ ∣ π ∣ 1/2 ≤ M ( 2 − b ) /3 ∣ π ∣ 1/2 (60) proves the first inequality in (91) ; the mesh condition gives the second.
In the Young case, write
g M , π : = ζ M , Λ − ζ M , Λ , π , a M , π : = M ( 2 − b ) / 4 ∣ π ∣ 1 / 2 . g^{M,\pi}:=\zeta^{M,\Lambda}-\zeta^{M,\Lambda,\pi},\qquad a_{M,\pi}:=M^{(2-b)/4}|\pi|^{1/2}. g M , π := ζ M , Λ − ζ M , Λ , π , a M , π := M ( 2 − b ) /4 ∣ π ∣ 1/2 . The mesh condition M ( 2 − b ) / 2 ∣ π ∣ ≤ 1 M^{(2-b)/2}|\pi|\leq1 M ( 2 − b ) /2 ∣ π ∣ ≤ 1 Λ ∣ π ∣ ≤ 1 \Lambda|\pi|\leq1 Λ∣ π ∣ ≤ 1 a M , π ≤ 1 a_{M,\pi}\leq1 a M , π ≤ 1 0 < b ≤ 1 0<b\leq1 0 < b ≤ 1 Proposition 4.10 therefore gives
( E ∥ g M , π ∥ C η ( [ 0 , T ] ) p ) 1 / p ≤ C a M , π . (94) \left(\mathbb{E}\|g^{M,\pi}\|_{C^\eta([0,T])}^p\right)^{1/p}\leq Ca_{M,\pi}. \tag*{(94)} ( E ∥ g M , π ∥ C η ([ 0 , T ]) p ) 1/ p ≤ C a M , π . (94) Moreover, Minkowski’s inequality applied to the time integral of Y M , Λ − Y η π ( ⋅ ) M , Λ , π Y^{M,\Lambda}-Y_{\eta_\pi(\cdot)}^{M,\Lambda,\pi} Y M , Λ − Y η π ( ⋅ ) M , Λ , π p = 2 p=2 p = 2
∥ g t M , π − g s M , π ∥ L 2 ( Ω ) ≤ C a M , π ∣ t − s ∣ . \|g_t^{M,\pi}-g_s^{M,\pi}\|_{L^2(\Omega)}\leq Ca_{M,\pi}|t-s|. ∥ g t M , π − g s M , π ∥ L 2 ( Ω ) ≤ C a M , π ∣ t − s ∣. Fix H 0 ∈ ( η , 1 ) H_0\in(\eta,1) H 0 ∈ ( η , 1 ) a M , π ≤ 1 a_{M,\pi}\leq1 a M , π ≤ 1 ∣ t − s ∣ ≤ T 1 − H 0 ∣ t − s ∣ H 0 |t-s|\leq T^{1-H_0}|t-s|^{H_0} ∣ t − s ∣ ≤ T 1 − H 0 ∣ t − s ∣ H 0 ∥ g M , π ∥ C η ( [ 0 , T ] ) 2 \|g^{M,\pi}\|_{C^\eta([0,T])^2} ∥ g M , π ∥ C η ([ 0 , T ] ) 2 M M M π \pi π (75) and Hölder’s inequality gives, for some c 0 > 0 c_0>0 c 0 > 0
sup M , π E exp ( c 0 ∥ ζ M , Λ , π ∥ C η ( [ 0 , T ] ) 2 ) < ∞ . \sup_{M,\pi}\mathbb{E}\exp\left(c_0\|\zeta^{M,\Lambda,\pi}\|_{C^\eta([0,T])}^2\right)<\infty. M , π sup E exp ( c 0 ∥ ζ M , Λ , π ∥ C η ([ 0 , T ]) 2 ) < ∞. Since 1 / η < 2 1/\eta< 2 1/ η < 2
sup M , π E exp ( c ∥ ζ M , Λ , π ∥ C n η ( [ 0 , T ] ) 1 / η ) < ∞ \sup_{M,\pi} \mathbb{E}\exp\left(c\|\zeta^{M,\Lambda,\pi}\|_{C^{\eta}_{n}([0,T])}^{1/\eta}\right)<\infty M , π sup E exp ( c ∥ ζ M , Λ , π ∥ C n η ([ 0 , T ]) 1/ η ) < ∞ for every fixed c > 0 c>0 c > 0
Let X ~ M , Λ , π \widetilde{X}^{M,\Lambda,\pi} X M , Λ , π ζ M , Λ , π \zeta^{M,\Lambda,\pi} ζ M , Λ , π x 0 x_0 x 0
∥ X − X M , Λ , π ∥ ∞ ≤ ∥ X − X M , Λ ∥ ∞ + ∥ X M , Λ − X ~ M , Λ , π ∥ ∞ + ∥ X ~ M , Λ , π − X M , Λ , π ∥ ∞ . \|X-X^{M,\Lambda,\pi}\|_{\infty}\leq\|X-X^{M,\Lambda}\|_{\infty}+\|X^{M,\Lambda}-\widetilde{X}^{M,\Lambda,\pi}\|_{\infty}+\|\widetilde{X}^{M,\Lambda,\pi}-X^{M,\Lambda,\pi}\|_{\infty}. ∥ X − X M , Λ , π ∥ ∞ ≤ ∥ X − X M , Λ ∥ ∞ + ∥ X M , Λ − X M , Λ , π ∥ ∞ + ∥ X M , Λ , π − X M , Λ , π ∥ ∞ . The first term is the spectral error already estimated in (74). We estimate the other two terms separately.
Lemma 4.5, the preceding exponential bounds, (93) at moment order 2 p 2p 2 p
( E sup t ≤ T ∣ X t M , Λ − X ~ t M , Λ , π ∣ p ) 1 / p ≤ C a M , π . \left(\mathbb{E}\sup_{t\leq T}\left|X^{M,\Lambda}_{t}-\widetilde{X}^{M,\Lambda,\pi}_{t}\right|^{p}\right)^{1/p}\leq Ca_{M,\pi}. ( E t ≤ T sup X t M , Λ − X t M , Λ , π p ) 1/ p ≤ C a M , π . Since ζ M , Λ , π \zeta^{M,\Lambda,\pi} ζ M , Λ , π Z π : = Y η π ( ⋅ ) M , Λ , π Z^{\pi}:=Y^{M,\Lambda,\pi}_{\eta_{\pi}(\cdot)} Z π := Y η π ( ⋅ ) M , Λ , π
( E sup t ≤ T ∣ X ~ t M , Λ , π − X t M , Λ , π ∣ p ) 1 / p ≤ C ∣ π ∣ 2 η − 1 . \left(\mathbb{E}\sup_{t\leq T}\left|\widetilde{X}^{M,\Lambda,\pi}_{t}-X^{M,\Lambda,\pi}_{t}\right|^{p}\right)^{1/p}\leq C|\pi|^{2\eta-1}. ( E t ≤ T sup X t M , Λ , π − X t M , Λ , π p ) 1/ p ≤ C ∣ π ∣ 2 η − 1 . The triangle inequality and (74) now prove (91).
For 1 < b < 2 1<b<2 1 < b < 2 ∣ π M ∣ ≤ M − 2 / 3 |\pi_M|\leq M^{-2/3} ∣ π M ∣ ≤ M − 2/3 0 < b ≤ 1 0<b\leq1 0 < b ≤ 1 M ( 2 − b ) / 2 ∣ π M ∣ → 0 M^{(2-b)/2}|\pi_M|\to0 M ( 2 − b ) /2 ∣ π M ∣ → 0 κ b , η , η ′ \kappa_{b,\eta,\eta'} κ b , η , η ′
Corollary 4.14 Under the assumptions of Theorem 4.13, let π = π M \pi=\pi_M π = π M
If 1 < b < 2 1<b<2 1 < b < 2 ∣ π M ∣ ≤ M − 2 / 3 |\pi_M|\leq M^{-2/3} ∣ π M ∣ ≤ M − 2/3
( E sup t ≤ T ∣ X t − X t M , M 2 / 3 , π M ∣ p ) 1 / p ≤ C M − ( b − 1 ) / 3 . \left(\mathbb{E}\sup_{t\leq T}\left|X_t-X^{M,M^{2/3},\pi_M}_t\right|^p\right)^{1/p}\leq CM^{-(b-1)/3}. ( E t ≤ T sup X t − X t M , M 2/3 , π M p ) 1/ p ≤ C M − ( b − 1 ) /3 . If 0 < b ≤ 1 0<b\leq1 0 < b ≤ 1 1 / 2 < η < η ′ < H b ∘ 1/2<\eta<\eta'<H_b^\circ 1/2 < η < η ′ < H b ∘
r b , η , η ′ : = b 4 ( 1 − η η ′ ) r_{b,\eta,\eta'}:=\frac{b}{4}\left(1-\frac{\eta}{\eta'}\right) r b , η , η ′ := 4 b ( 1 − η ′ η ) and
κ b , η , η ′ : = max { 2 − b 2 + 2 r b , η , η ′ , r b , η , η ′ 2 η − 1 } . (95) \kappa_{b,\eta,\eta'}:=\max\left\{\frac{2-b}{2}+2r_{b,\eta,\eta'},\frac{r_{b,\eta,\eta'}}{2\eta-1}\right\}. \tag*{(95)} κ b , η , η ′ := max { 2 2 − b + 2 r b , η , η ′ , 2 η − 1 r b , η , η ′ } . (95) If ∣ π M ∣ ≤ M − κ b , η , η ′ |\pi_M| \le M^{-\kappa_{b,\eta,\eta'}} ∣ π M ∣ ≤ M − κ b , η , η ′
( E sup t ≤ T ∣ X t − X t M , M 1 / 2 , π M ∣ p ) 1 / p ≤ C M − r b , η , η ′ ( log ( 2 + M ) ) ( 1 − η / η ′ ) / 2 . \left(\mathbb{E}\sup_{t\le T}\left|X_t-X_t^{M,M^{1/2},\pi_M}\right|^p\right)^{1/p}
\le C M^{-r_{b,\eta,\eta'}}\left(\log(2+M)\right)^{(1-\eta/\eta')/2}. ( E t ≤ T sup X t − X t M , M 1/2 , π M p ) 1/ p ≤ C M − r b , η , η ′ ( log ( 2 + M ) ) ( 1 − η / η ′ ) /2 . These mesh choices preserve the spectral convergence rate. The choice b = 1 b=1 b = 1 η = 0.7 \eta=0.7 η = 0.7 η ′ = 0.9 \eta'=0.9 η ′ = 0.9
r 1 , 0.7 , 0.9 = 1 18 , κ 1 , 0.7 , 0.9 = 11 18 . r_{1,0.7,0.9}=\frac{1}{18},\qquad\kappa_{1,0.7,0.9}=\frac{11}{18}. r 1 , 0.7 , 0.9 = 18 1 , κ 1 , 0.7 , 0.9 = 18 11 . Proof. The finite-variation statement is (90). In the Young regime, (94) gives
κ b , η , η ′ 2 − 2 − b 4 ≥ r b , η , η ′ , κ b , η , η ′ ( 2 η − 1 ) ≥ r b , η , η ′ . \frac{\kappa_{b,\eta,\eta'}}{2}-\frac{2-b}{4}\ge r_{b,\eta,\eta'},\qquad\kappa_{b,\eta,\eta'}(2\eta-1)\ge r_{b,\eta,\eta'}. 2 κ b , η , η ′ − 4 2 − b ≥ r b , η , η ′ , κ b , η , η ′ ( 2 η − 1 ) ≥ r b , η , η ′ . The first inequality also implies the mesh restriction in Theorem 4.13. Substitution in (91) proves the claim. □
Remark 4.15 The convergence estimates depend only on the maximal temporal cell width. To relate the computational cost to that width, we also bound the number of time steps. Call the temporal meshes quasi-uniform if there is a fixed c > 0 c>0 c > 0 c ∣ π M ∣ ≤ Δ n ≤ ∣ π M ∣ c|\pi_M|\le\Delta_n\le|\pi_M| c ∣ π M ∣ ≤ Δ n ≤ ∣ π M ∣ M M M N M N_M N M π M \pi_M π M a M ≍ b M a_M\asymp b_M a M ≍ b M c b M ≤ a M ≤ C b M c b_M\le a_M\le C b_M c b M ≤ a M ≤ C b M M M M 0 < c ≤ C < ∞ 0<c\le C<\infty 0 < c ≤ C < ∞ M M M ∣ π M ∣ ≍ M − 2 / 3 |\pi_M|\asymp M^{-2/3} ∣ π M ∣ ≍ M − 2/3 N M ≍ M 2 / 3 N_M\asymp M^{2/3} N M ≍ M 2/3 M M M O ( M N M ) = O ( M 5 / 3 ) O(MN_M)=O(M^{5/3}) O ( M N M ) = O ( M 5/3 ) ∣ π M ∣ ≍ M − κ b , η , η ′ |\pi_M|\asymp M^{-\kappa_{b,\eta,\eta'}} ∣ π M ∣ ≍ M − κ b , η , η ′ N M ≍ M κ b , η , η ′ N_M\asymp M^{\kappa_{b,\eta,\eta'}} N M ≍ M κ b , η , η ′ O ( M 1 + κ b , η , η ′ ) O(M^{1+\kappa_{b,\eta,\eta'}}) O ( M 1 + κ b , η , η ′ ) v f , n v_{f,n} v f , n
Taking η ↓ 1 / 2 \eta\downarrow1/2 η ↓ 1/2 η ′ \eta' η ′ r b , η , η ′ r_{b,\eta,\eta'} r b , η , η ′ r b , η , η ′ / ( 2 η − 1 ) r_{b,\eta,\eta'}/(2\eta-1) r b , η , η ′ / ( 2 η − 1 ) η , η ′ \eta,\eta' η , η ′ M M M η = η ( M ) \eta=\eta(M) η = η ( M )
Statements and Declarations
Funding. J. H. Ramírez-González acknowledges support from the São Paulo Research Foundation (FAPESP), grant 2025/03804-0.
Competing interests. The authors have no relevant financial or non-financial interests to disclose.
Data availability. Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
Use of AI assistance. AI tools were used to improve the wording and flow of the manuscript.
