Introduction
Dry (Coulomb) friction governs stick–slip oscillations and the positioning accuracy of machines [1 , 2 ] . While a body slides, the friction force has magnitude Δ > 0 \Delta> 0 Δ > 0 [ − Δ , Δ ] [-\Delta,\Delta] [ − Δ , Δ ] φ ( v ) = Δ ∣ v ∣ \varphi(v) = \Delta|v| φ ( v ) = Δ∣ v ∣ ∂ φ ( v ) = { Δ sgn ( v ) } \partial\varphi(v) = \{\Delta\operatorname{sgn}(v)\} ∂ φ ( v ) = { Δ sgn ( v )} v ≠ 0 v \ne0 v = 0 ∂ φ ( 0 ) = [ − Δ , Δ ] \partial\varphi(0) = [-\Delta,\Delta] ∂ φ ( 0 ) = [ − Δ , Δ ] [3 , 4 ] , well posed by maximal monotone operator theory [5 , 6 ] . Random loads lead to stochastic differential inclusions [6 , 7 , 8 ] .
Under Gaussian white noise, dry friction is well understood. De Gennes [9 ] studied the stationary velocity law and the velocity correlations. The transition density is known in closed form for pure dry friction [10 , 11 , 12 ] . Touchette, Van der Straeten and Just [12 ] solved the Fokker–Planck equation by spectral series when viscous friction and a constant force are added, and Touchette, Prellberg and Just [13 ] obtained the Laplace transform of the propagator in closed form. See also [14 , 15 , 16 ] and the experiments [17 , 18 , 19 ] . Physical forcing, however, is colored [20 , 21 ] . A finite correlation time changes the dynamics qualitatively. It induces genuine sticking [22 , 23 ] , leads to a piecewise-deterministic description with a critical correlation time [24 , 25 ] , and can make the white-noise approximation inaccurate for statistics such as the diffusivity [26 ] . Thus white-noise results cannot replace computations at positive correlation time.
The forcing can be represented by a linear functional of a fast multivariate Ornstein–Uhlenbeck (OU) process [27 ] . Let ε 2 \varepsilon^2 ε 2 ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 [28 , 29 , 30 ] , with the Wong–Zakai interpretation of the limit [31 ] . For fixed ε \varepsilon ε ε − 2 \varepsilon^{-2} ε − 2 [32 , 33 , 34 , 35 ] reduce this cost when a correlated quantity with a computable mean is available. Garnier and Mertz Garnier and Mertz [27 ] drive the colored-noise system and its white-noise limit by the same Wiener process. They compute the expectation of the limit from its Kolmogorov equation, which has no large terms, and estimate only the small difference by Monte Carlo. For smooth observables and the systems considered here, the asymptotic variance is O ( ε 2 ) O(\varepsilon^2) O ( ε 2 ) [27 ] . Other variance reduction methods require structure that is not natural here. Importance sampling [36 ] needs a biasing measure, subset simulation [37 ] intermediate events, and multilevel Monte Carlo [38 , 39 , 40 ] a hierarchy of resolutions along which the observable varies smoothly.
Reliability analysis, however, needs non-smooth observables [41 , 42 , 43 ] . Examples are the indicator 1 { ∣ v ∣ ≥ v f } \mathbf{1}_{\{|v|\ge v_f\}} 1 { ∣ v ∣ ≥ v f } [27 ] indicate a variance of order ε \varepsilon ε ε 2 \varepsilon^2 ε 2 [27 ] requires observables with bounded derivatives. For an indicator, the difference along the coupled trajectories is not small but rarely nonzero. Its second moment is essentially the probability that the threshold lies between the two terminal velocities. Bounding this probability requires regularity of the white-noise law near the threshold and tail bounds for the coupling error. The mean-square coupling bound of [27 ] alone cannot give a better rate than ε 2 / 3 \varepsilon^{2/3} ε 2/3 f f f [44 ] bounds the L q L^q L q f f f ε 2 / 3 \varepsilon^{2/3} ε 2/3 [45 , 46 ] . We obtain sharper rates for the inertial dry-friction system
X ¨ ε ( t ) + ∂ φ ( X ˙ ε ( t ) ) ∋ b ( X ε ( t ) , X ˙ ε ( t ) ) + σ ξ ε ( t ) , ( X ε ( 0 ) , X ˙ ε ( 0 ) ) = ( x 0 , v 0 ) , t ∈ [ 0 , T ] . (1) \ddot{X}^{\varepsilon}(t)+\partial\varphi\left(\dot{X}^{\varepsilon}(t)\right)\ni b\left(X^{\varepsilon}(t),\dot{X}^{\varepsilon}(t)\right)+\sigma\xi^{\varepsilon}(t),\qquad\left(X^{\varepsilon}(0),\dot{X}^{\varepsilon}(0)\right)=(x_0,v_0),\qquad t\in[0,T]. \tag*{(1)} X ¨ ε ( t ) + ∂ φ ( X ˙ ε ( t ) ) ∋ b ( X ε ( t ) , X ˙ ε ( t ) ) + σ ξ ε ( t ) , ( X ε ( 0 ) , X ˙ ε ( 0 ) ) = ( x 0 , v 0 ) , t ∈ [ 0 , T ] . (1) where ξ ε \xi^\varepsilon ξ ε ε 2 \varepsilon^2 ε 2 σ > 0 \sigma>0 σ > 0 b b b [12 ] : b = 0 b=0 b = 0 b ( x , v ) = − γ v b(x,v)=-\gamma v b ( x , v ) = − γ v γ > 0 \gamma>0 γ > 0 b ( x , v ) = − γ v + ℓ b(x,v)=-\gamma v+\ell b ( x , v ) = − γ v + ℓ γ > 0 \gamma>0 γ > 0 ℓ ∈ R \ell\in\mathbb{R} ℓ ∈ R b b b V ε = X ˙ ε V^\varepsilon=\dot{X}^\varepsilon V ε = X ˙ ε [27 ] . This scalar structure is exploited in our analysis.
Contributions
Throughout, V ε V^\varepsilon V ε V 0 V^0 V 0
Main result. Theorem 3.1 bounds the second moment E [ ( f ( V ε ( T ) ) − f ( V 0 ( T ) ) ) 2 ] \mathbb{E}[(f(V^\varepsilon(T))-f(V^0(T)))^2] E [( f ( V ε ( T )) − f ( V 0 ( T )) ) 2 ] [27 ] . The bound holds for four classes of observables. Up to logarithmic factors in the last three cases,
E [ ( f ( V ε ( T ) ) − f ( V 0 ( T ) ) ) 2 ] ≲ { ε 2 α , f α -H o ¨ lder , ε , f of bounded variation , ε 1 + α , f α -H o ¨ lder and of bounded variation , ε min { 2 , 1 + 2 α } , f with finitely many α -cusps . \mathbb{E}\left[(f(V^\varepsilon(T))-f(V^0(T)))^2\right]\lesssim
\begin{cases}
\varepsilon^{2\alpha}, & f\ \alpha\text{-Hölder},\\
\varepsilon, & f\ \text{of bounded variation},\\
\varepsilon^{1+\alpha}, & f\ \alpha\text{-Hölder and of bounded variation},\\
\varepsilon^{\min\{2,1+2\alpha\}}, & f\ \text{with finitely many }\alpha\text{-cusps}.
\end{cases} E [ ( f ( V ε ( T )) − f ( V 0 ( T )) ) 2 ] ≲ ⎩ ⎨ ⎧ ε 2 α , ε , ε 1 + α , ε m i n { 2 , 1 + 2 α } , f α -H o ¨ lder , f of bounded variation , f α -H o ¨ lder and of bounded variation , f with finitely many α -cusps . where observables of bounded variation satisfy a mild one-sided continuity condition. The first rate extends the rate ε 2 \varepsilon^2 ε 2 [27 ] from smooth to Lipschitz observables (α = 1 \alpha= 1 α = 1 ε 2 / 3 \varepsilon^{2/3} ε 2/3 ε \varepsilon ε ε 2 \varepsilon^2 ε 2 α > 1 / 2 \alpha> 1/2 α > 1/2 [27 ] .
The proof uses two structural properties. First, a pathwise comparison principle shows that the solution map from integrated forcing to velocity is 2-Lipschitz for the uniform norm (Proposition B.1); its continuity is known [6 Proposition 4.16] . The integrated colored and white forcings differ by an explicit functional of the OU process of order ε \varepsilon ε ϑ ε \vartheta_\varepsilon ϑ ε Proposition B.2 ). Second, the law of the white-noise terminal velocity has a density with an explicit bound in all three regimes (Proposition B.3 ). It is obtained by comparing the white-noise velocity, through Girsanov’s theorem, with pure dry friction, whose transition density is known in closed form [11 ] . In regimes (II) and (III), the propagator is known through spectral series [12 ] and through its Laplace transform [13 ] , which give no explicit bound; see Remark B.1 for general density estimates.
Consequences. The rates imply bounds on the bias of the white-noise approximation, on variance reduction, and on sample size (Corollaries 3.1 to 3.3).
Numerical experiments. The coupled time discretization satisfies a discrete comparison principle (Proposition 5.1 ). Thus the coupling survives discretization. Discretization affects the bias of the estimators but not the coupling error, so strong rates for such schemes [47 ] are not needed. In the three regimes, we test the predicted exponents, the coupling-error bounds and the variance reduction (§ 5 ).
Organization and notation
The paper is organized as follows. Section 2 sets up the model and estimators. Section 3 states the main results. Section 4 contains the proofs. Section 5 reports numerical experiments. Section 6 concludes. Appendix A contains auxiliary lemmas, Appendix B the comparison principle, the coupling error and the density bound, and Appendix C implementation details.
We write a ∧ b = min { a , b } a \wedge b = \min\{a,b\} a ∧ b = min { a , b } a ∨ b = max { a , b } a \vee b = \max\{a,b\} a ∨ b = max { a , b } x + = max { x , 0 } x_+ = \max\{x,0\} x + = max { x , 0 } sgn ( x ) = x / ∣ x ∣ \operatorname{sgn}(x) = x/\lvert x\rvert sgn ( x ) = x / ∣ x ∣ x ≠ 0 x \ne0 x = 0 sgn ( 0 ) = 0 \operatorname{sgn}(0) = 0 sgn ( 0 ) = 0 ⟦ x , y ⟧ ≔ [ x ∧ y , x ∨ y ] \llbracket x,y\rrbracket\coloneqq[x \wedge y,x \vee y] [ [ x , y ] ] : = [ x ∧ y , x ∨ y ] x , y ∈ R x,y \in\mathbb{R} x , y ∈ R
We denote by ∣ ⋅ ∣ \lvert\cdot\rvert ∣ ⋅ ∣ ∥ ⋅ ∥ \lVert\cdot\rVert ∥ ⋅ ∥ Leb \operatorname{Leb} Leb Φ ‾ ( x ) ≔ ( 2 π ) − 1 / 2 ∫ x ∞ e − z 2 / 2 d z \overline{\Phi}(x) \coloneqq(2\pi)^{-1/2}\int_x^\infty e^{-z^2/2}\,\mathrm{d}z Φ ( x ) : = ( 2 π ) − 1/2 ∫ x ∞ e − z 2 /2 d z C 0 0 ( [ 0 , T ] ) C^0_0([0,T]) C 0 0 ([ 0 , T ]) w : [ 0 , T ] → R w\colon[0,T] \to\mathbb{R} w : [ 0 , T ] → R w ( 0 ) = 0 w(0)=0 w ( 0 ) = 0 ∥ w ∥ ∞ = sup t ≤ T ∣ w ( t ) ∣ \lVert w\rVert_\infty= \sup_{t\le T}\lvert w(t)\rvert ∥ w ∥ ∞ = sup t ≤ T ∣ w ( t )∣ C C C
Model and mathematical preliminaries
Throughout, T > 0 T > 0 T > 0 x 0 , v 0 ∈ R x_0,v_0 \in\mathbb{R} x 0 , v 0 ∈ R Δ > 0 \Delta> 0 Δ > 0 σ > 0 \sigma> 0 σ > 0 φ ( v ) = Δ ∣ v ∣ \varphi(v) = \Delta\lvert v\rvert φ ( v ) = Δ ∣ v ∣
The dry-friction inclusion
Assumption A1 (velocity-dependent drift). The drift in (1.1) is b ( x , v ) = b ‾ ( v ) ≔ − γ v + ℓ b(x,v)=\overline{b}(v) \coloneqq-\gamma v+\ell b ( x , v ) = b ( v ) : = − γ v + ℓ γ = ℓ = 0 \gamma=\ell=0 γ = ℓ = 0 γ > 0 \gamma>0 γ > 0 ℓ = 0 \ell=0 ℓ = 0 γ > 0 \gamma>0 γ > 0 ℓ ∈ R \ell\in\mathbb{R} ℓ ∈ R
Under Assumption A1, system (1.1) is equivalent to
X ˙ ε = V ε , V ˙ ε + ∂ φ ( V ε ) ∋ b ‾ ( V ε ) + σ ξ ε ( t ) , ( X ε ( 0 ) , V ε ( 0 ) ) = ( x 0 , v 0 ) , t ∈ [ 0 , T ] . (2) \dot{X}^{\varepsilon}=V^{\varepsilon},\qquad
\dot{V}^{\varepsilon}+\partial\varphi(V^{\varepsilon})\ni\overline{b}(V^{\varepsilon})+\sigma\xi^{\varepsilon}(t),\qquad
(X^{\varepsilon}(0),V^{\varepsilon}(0))=(x_0,v_0),\qquad t\in[0,T]. \tag*{(2)} X ˙ ε = V ε , V ˙ ε + ∂ φ ( V ε ) ∋ b ( V ε ) + σ ξ ε ( t ) , ( X ε ( 0 ) , V ε ( 0 )) = ( x 0 , v 0 ) , t ∈ [ 0 , T ] . (2) The velocity inclusion is autonomous and the position is recovered by quadrature. We want to compute E [ f ( V ε ( T ) ) ] \mathbb{E}[f(V^\varepsilon(T))] E [ f ( V ε ( T ))] f f f
U ( v ) ≔ γ 2 v 2 − ℓ v + Δ ∣ v ∣ , U ∗ ≔ inf v ∈ R U ( v ) = { 0 , regime (I) , − ( ( ∣ ℓ ∣ − Δ ) + ) 2 2 γ , regimes (II)–(III) , (3) U(v) \coloneqq\frac{\gamma}{2}v^2-\ell v+\Delta\lvert v\rvert,\qquad U_\ast\coloneqq\inf_{v\in\mathbb{R}} U(v)=\begin{cases} 0, & \text{regime (I)}, \\ -\frac{((\lvert\ell\rvert-\Delta)_+)^2}{2\gamma}, & \text{regimes (II)--(III)}, \end{cases} \tag*{(3)} U ( v ) : = 2 γ v 2 − ℓ v + Δ ∣ v ∣ , U ∗ : = v ∈ R inf U ( v ) = { 0 , − 2 γ ((∣ ℓ ∣ − Δ ) + ) 2 , regime (I) , regimes (II)–(III) , (3) so that b ˉ − ∂ φ = − ∂ U \bar b-\partial\varphi=-\partial U b ˉ − ∂ φ = − ∂ U U U U ∣ ℓ ∣ < Δ \lvert\ell\rvert<\Delta ∣ ℓ ∣ < Δ U U U 0 0 0 ∣ ℓ ∣ > Δ \lvert\ell\rvert>\Delta ∣ ℓ ∣ > Δ
Colored noise and coupling
Assumption A2 (colored noise). Let d , d ′ ≥ 1 d,d'\ge1 d , d ′ ≥ 1 A ∈ R d × d \mathbf{A}\in\mathbb{R}^{d\times d} A ∈ R d × d K ∈ R d × d ′ \mathbf{K}\in\mathbb{R}^{d\times d'} K ∈ R d × d ′ r ∈ R d ∖ { 0 } \mathbf{r}\in\mathbb{R}^{d}\setminus\{0\} r ∈ R d ∖ { 0 } ( Ω , F , ( F t ) t ≥ 0 , P ) (\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P}) ( Ω , F , ( F t ) t ≥ 0 , P ) d ′ d' d ′ ( F t ) (\mathcal{F}_t) ( F t ) W \mathbf{W} W F 0 \mathcal{F}_0 F 0 η 0 ∼ N ( 0 , C ) \boldsymbol{\eta}_0\sim\mathcal{N}(0,\mathbf{C}) η 0 ∼ N ( 0 , C )
C ≔ ∫ 0 ∞ e − A u K K T e − A T u d u \mathbf{C}\coloneqq\int_0^\infty e^{-\mathbf{A}u}\mathbf{K}\mathbf{K}^{\mathsf{T}}e^{-\mathbf{A}^{\mathsf{T}}u}\,du C : = ∫ 0 ∞ e − A u K K T e − A T u d u is the unique solution of A C + C A T = K K T \mathbf{A}\mathbf{C}+\mathbf{C}\mathbf{A}^{\mathsf{T}}=\mathbf{K}\mathbf{K}^{\mathsf{T}} AC + C A T = K K T ε > 0 \varepsilon>0 ε > 0 η ε \boldsymbol{\eta}^{\varepsilon} η ε
d η ε ( t ) = − A ε 2 η ε ( t ) d t + K ε d W ( t ) , η ε ( 0 ) = η 0 , (4) d\boldsymbol{\eta}^{\varepsilon}(t)=-\frac{\mathbf{A}}{\varepsilon^2}\boldsymbol{\eta}^{\varepsilon}(t)\,dt+\frac{\mathbf{K}}{\varepsilon}\,d\mathbf{W}(t),\qquad\boldsymbol{\eta}^{\varepsilon}(0)=\boldsymbol{\eta}_0, \tag*{(4)} d η ε ( t ) = − ε 2 A η ε ( t ) d t + ε K d W ( t ) , η ε ( 0 ) = η 0 , (4) and the colored noise is ξ ε ( t ) ≔ r ⋅ η ε ( t ) / ε \xi^{\varepsilon}(t)\coloneqq\mathbf{r}\cdot\boldsymbol{\eta}^{\varepsilon}(t)/\varepsilon ξ ε ( t ) : = r ⋅ η ε ( t ) / ε
The process η ε \boldsymbol{\eta}^{\varepsilon} η ε η ^ ( u ) ≔ η ε ( ε 2 u ) \widehat{\boldsymbol{\eta}}(u)\coloneqq\boldsymbol{\eta}^{\varepsilon}(\varepsilon^2u) η ( u ) : = η ε ( ε 2 u ) d η ^ = − A η ^ d u + K d W ^ d\widehat{\boldsymbol{\eta}}=-\mathbf{A}\widehat{\boldsymbol{\eta}}\,du+\mathbf{K}\,d\widehat{\mathbf{W}} d η = − A η d u + K d W W ^ ( u ) ≔ ε − 1 W ( ε 2 u ) \widehat{\mathbf{W}}(u)\coloneqq\varepsilon^{-1}\mathbf{W}(\varepsilon^2u) W ( u ) : = ε − 1 W ( ε 2 u ) ξ ε \xi^{\varepsilon} ξ ε O ( ε − 1 ) O(\varepsilon^{-1}) O ( ε − 1 ) O ( ε 2 ) O(\varepsilon^2) O ( ε 2 ) d ≥ 2 d\ge2 d ≥ 2 [20 ] , [27 Examples 3.1–3.4] . Since A \mathbf{A} A (4) and multiplying on the left by ε r T A − 1 \varepsilon\mathbf{r}^{\mathsf{T}}\mathbf{A}^{-1} ε r T A − 1 t ≥ 0 t\ge0 t ≥ 0
∫ 0 t ξ ε ( u ) d u = r T A − 1 K W ( t ) − ε r T A − 1 ( η ε ( t ) − η 0 ) . (5) \int_0^t \xi^{\varepsilon}(u)\,du=\mathbf{r}^{\mathsf{T}}\mathbf{A}^{-1}\mathbf{K}\mathbf{W}(t)-\varepsilon\mathbf{r}^{\mathsf{T}}\mathbf{A}^{-1}\left(\boldsymbol{\eta}^{\varepsilon}(t)-\boldsymbol{\eta}_0\right). \tag*{(5)} ∫ 0 t ξ ε ( u ) d u = r T A − 1 KW ( t ) − ε r T A − 1 ( η ε ( t ) − η 0 ) . (5) We set c ≔ A − T r ∈ R d ∖ { 0 } \mathbf{c}\coloneqq\mathbf{A}^{-\mathsf{T}}\mathbf{r}\in\mathbb{R}^{d}\setminus\{0\} c : = A − T r ∈ R d ∖ { 0 } κ 0 ≔ σ ∣ c ∣ > 0 \kappa_0\coloneqq\sigma\lvert\mathbf{c}\rvert>0 κ 0 : = σ ∣ c ∣ > 0
N ε ( t ) ≔ σ ∫ 0 t ξ ε ( u ) d u , N 0 ( t ) ≔ σ c T K W ( t ) , h ε ( t ) ≔ N ε ( t ) − N 0 ( t ) = − ε σ c T ( η ε ( t ) − η 0 ) , N^{\varepsilon}(t)\coloneqq\sigma\int_0^t \xi^{\varepsilon}(u)\,du,\qquad N^0(t)\coloneqq\sigma\mathbf{c}^{\mathsf{T}}\mathbf{K}\mathbf{W}(t),\qquad h^{\varepsilon}(t)\coloneqq N^{\varepsilon}(t)-N^0(t)=-\varepsilon\sigma\mathbf{c}^{\mathsf{T}}\left(\boldsymbol{\eta}^{\varepsilon}(t)-\boldsymbol{\eta}_0\right), N ε ( t ) : = σ ∫ 0 t ξ ε ( u ) d u , N 0 ( t ) : = σ c T KW ( t ) , h ε ( t ) : = N ε ( t ) − N 0 ( t ) = − ε σ c T ( η ε ( t ) − η 0 ) , so that
sup u ≤ t ∣ h ε ( u ) ∣ ≤ 2 ε κ 0 sup u ≤ t ∣ η ε ( u ) ∣ , t ≥ 0. (6) \sup_{u\le t}\lvert h^{\varepsilon}(u)\rvert\le2\varepsilon\kappa_0\sup_{u\le t}\lvert\boldsymbol{\eta}^{\varepsilon}(u)\rvert,\qquad t\ge0. \tag*{(6)} u ≤ t sup ∣ h ε ( u )∣ ≤ 2 ε κ 0 u ≤ t sup ∣ η ε ( u )∣ , t ≥ 0. (6) The process N 0 N^0 N 0 ς 2 \varsigma^2 ς 2 ς ≔ σ ∣ K T c ∣ = σ ∣ K T A − T r ∣ \varsigma\coloneqq\sigma\lvert\mathbf{K}^{\mathsf{T}}\mathbf{c}\rvert=\sigma\lvert\mathbf{K}^{\mathsf{T}}\mathbf{A}^{-\mathsf{T}}\mathbf{r}\rvert ς : = σ ∣ K T c ∣ = σ ∣ K T A − T r ∣
Assumption A3 (non-degenerate white-noise limit). ς > 0 \varsigma>0 ς > 0
Under Assumption A3, B ≔ ( K T c ) ⋅ W / ∣ K T c ∣ B\coloneqq(\mathbf{K}^{\mathsf{T}}\mathbf{c})\cdot\mathbf{W}/\lvert\mathbf{K}^{\mathsf{T}}\mathbf{c}\rvert B : = ( K T c ) ⋅ W / ∣ K T c ∣ ( F t ) (\mathcal{F}_t) ( F t ) N 0 = ς B N^0=\varsigma B N 0 = ς B (6) and Lemma A.3, N ε → N 0 N^{\varepsilon}\to N^0 N ε → N 0 [ 0 , T ] [0,T] [ 0 , T ] L p L^p L p p > 0 p>0 p > 0
Solution concept and coupled pair
Since N 0 N^0 N 0
Definition 2.1 (Solution). Let w ∈ C 0 ( [ 0 , T ] ) w \in C^0([0,T]) w ∈ C 0 ([ 0 , T ])
d v ( t ) + ∂ φ ( v ( t ) ) d t ∋ b ˉ ( v ( t ) ) d t + d w ( t ) , v ( 0 ) = v 0 , (7) \mathrm{d}v(t) + \partial\varphi(v(t))\,\mathrm{d}t \ni\bar{b}(v(t))\,\mathrm{d}t + \mathrm{d}w(t), \qquad v(0)=v_0, \tag*{(7)} d v ( t ) + ∂ φ ( v ( t )) d t ∋ b ˉ ( v ( t )) d t + d w ( t ) , v ( 0 ) = v 0 , (7) is a pair ( v , k ) (v,k) ( v , k ) v ∈ C ( [ 0 , T ] ) v \in C([0,T]) v ∈ C ([ 0 , T ]) k : [ 0 , T ] → R k:[0,T]\to\mathbb{R} k : [ 0 , T ] → R k ( t ) ∈ ∂ φ ( v ( t ) ) k(t)\in\partial\varphi(v(t)) k ( t ) ∈ ∂ φ ( v ( t )) t t t
v ( t ) = v 0 + ∫ 0 t ( b ˉ ( v ( u ) ) − k ( u ) ) d u + w ( t ) , t ∈ [ 0 , T ] . (8) v(t)=v_0+\int_0^t\bigl(\bar{b}(v(u))-k(u)\bigr)\,\mathrm{d}u+w(t), \qquad t\in[0,T]. \tag*{(8)} v ( t ) = v 0 + ∫ 0 t ( b ˉ ( v ( u )) − k ( u ) ) d u + w ( t ) , t ∈ [ 0 , T ] . (8) Necessarily ∣ k ∣ ≤ Δ |k|\leq\Delta ∣ k ∣ ≤ Δ k k k
If w ∈ C 1 w\in C^1 w ∈ C 1 v v v v ˙ + ∂ φ ( v ) ∋ b ˉ ( v ) + w ˙ \dot{v}+\partial\varphi(v)\ni\bar{b}(v)+\dot{w} v ˙ + ∂ φ ( v ) ∋ b ˉ ( v ) + w ˙ φ \varphi φ (7) is a generalized Skorokhod problem, which is well posed by [6 Theorem 4.17] : for every w ∈ C 0 ( [ 0 , T ] ) w\in C^0([0,T]) w ∈ C 0 ([ 0 , T ]) ( v , k ) (v,k) ( v , k ) k k k v = S ( w ) v=S(w) v = S ( w ) S S S S ( w ) ∣ [ 0 , t ] S(w)|_{[0,t]} S ( w ) ∣ [ 0 , t ] w ∣ [ 0 , t ] w|_{[0,t]} w ∣ [ 0 , t ]
V ε ≜ S ( N ε ) , V 0 ≜ S ( N 0 ) , (9) V^\varepsilon\triangleq S(N^\varepsilon), \qquad V^0\triangleq S(N^0), \tag*{(9)} V ε ≜ S ( N ε ) , V 0 ≜ S ( N 0 ) , (9) with friction forces k ε k^\varepsilon k ε k 0 k^0 k 0 S S S V ε V^\varepsilon V ε (2) pathwise, V 0 V^0 V 0 d V 0 + ∂ φ ( V 0 ) d t ∋ b ˉ ( V 0 ) d t + d N 0 ( t ) \mathrm{d}V^0+\partial\varphi(V^0)\,\mathrm{d}t\ni\bar{b}(V^0)\,\mathrm{d}t+\mathrm{d}N^0(t) d V 0 + ∂ φ ( V 0 ) d t ∋ b ˉ ( V 0 ) d t + d N 0 ( t ) (9) is the coupling of [27 ] . We call Z ε ( t ) ≜ V ε ( t ) − V 0 ( t ) Z^\varepsilon(t)\triangleq V^\varepsilon(t)-V^0(t) Z ε ( t ) ≜ V ε ( t ) − V 0 ( t ) t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ] V ε ≡ 0 V^\varepsilon\equiv0 V ε ≡ 0 k ε = ℓ + σ ξ ε ∈ [ − Δ , Δ ] k^\varepsilon=\ell+\sigma\xi^\varepsilon\in[-\Delta,\Delta] k ε = ℓ + σ ξ ε ∈ [ − Δ , Δ ] Lemma A.1 ).
Classes of observables
Definition 2.2 (Observables). Let f : R → R f:\mathbb{R}\to\mathbb{R} f : R → R
(a) Hölder class. For α ∈ ( 0 , 1 ] \alpha\in(0,1] α ∈ ( 0 , 1 ] [ f ] α ≜ sup x ≠ y ∣ f ( x ) − f ( y ) ∣ / ∣ x − y ∣ α [f]_\alpha\triangleq\sup_{x\ne y}|f(x)-f(y)|/|x-y|^\alpha [ f ] α ≜ sup x = y ∣ f ( x ) − f ( y ) ∣/∣ x − y ∣ α C 0 , α ≜ { f : [ f ] α < ∞ } C^{0,\alpha}\triangleq\{f:[f]_\alpha<\infty\} C 0 , α ≜ { f : [ f ] α < ∞ } C 0 , α C^{0,\alpha} C 0 , α
(b) Bounded variation. Define TV ( f ) ≜ sup ∑ i = 1 n ∣ f ( x i ) − f ( x i − 1 ) ∣ \operatorname{TV}(f)\triangleq\sup\sum_{i=1}^n|f(x_i)-f(x_{i-1})| TV ( f ) ≜ sup ∑ i = 1 n ∣ f ( x i ) − f ( x i − 1 ) ∣ n ≥ 1 n\geq1 n ≥ 1 − ∞ < x 0 < x 1 < ⋯ < x n < ∞ -\infty<x_0<x_1<\cdots<x_n<\infty − ∞ < x 0 < x 1 < ⋯ < x n < ∞ BV ≜ { f : TV ( f ) < ∞ } \operatorname{BV}\triangleq\{f:\operatorname{TV}(f)<\infty\} BV ≜ { f : TV ( f ) < ∞ }
(c) Condition (J). For f ∈ BV f\in\operatorname{BV} f ∈ BV f ( x − ) = lim y ↑ x f ( y ) f(x-)=\lim_{y\uparrow x}f(y) f ( x − ) = lim y ↑ x f ( y ) f ( x + ) = lim y ↓ x f ( y ) f(x+)=\lim_{y\downarrow x}f(y) f ( x + ) = lim y ↓ x f ( y ) x ∈ R x\in\mathbb{R} x ∈ R f ∈ BV f\in\operatorname{BV} f ∈ BV f ( x ) ∈ { f ( x − ) , f ( x + ) } f(x)\in\{f(x-),f(x+)\} f ( x ) ∈ { f ( x − ) , f ( x + )} x ∈ R x\in\mathbb{R} x ∈ R BV J ≜ { f ∈ BV : f satisfies (J) } \operatorname{BV}_{\mathrm{J}}\triangleq\{f\in\operatorname{BV}:f\text{ satisfies (J)}\} BV J ≜ { f ∈ BV : f satisfies (J) }
(d) Cusp class. Fix a finite nonempty set S ⊂ R S\subset\mathbb{R} S ⊂ R α ∈ ( 0 , 1 ] \alpha\in(0,1] α ∈ ( 0 , 1 ] AC l o c ( R ) \operatorname{AC}_{\mathrm{loc}}(\mathbb{R}) AC loc ( R ) f ∈ AC l o c ( R ) f\in\operatorname{AC}_{\mathrm{loc}}(\mathbb{R}) f ∈ AC loc ( R ) [ f ] α , S ≜ ess sup x ∈ R ∖ S dist ( x , S ) 1 − α ∣ f ′ ( x ) ∣ [f]_{\alpha,S}\triangleq\operatorname*{ess\,sup}_{x\in\mathbb{R}\setminus S}\operatorname{dist}(x,S)^{1-\alpha}|f'(x)| [ f ] α , S ≜ ess sup x ∈ R ∖ S dist ( x , S ) 1 − α ∣ f ′ ( x ) ∣ f ′ f' f ′ f f f
K S α ≜ { f ∈ AC l o c ( R ) : [ f ] α , S < ∞ } . \mathcal{K}^{\alpha}_{S}\triangleq\{f\in\operatorname{AC}_{\mathrm{loc}}(\mathbb{R}):[f]_{\alpha,S}<\infty\}. K S α ≜ { f ∈ AC loc ( R ) : [ f ] α , S < ∞ } . For f ∈ BV f\in\operatorname{BV} f ∈ BV f f f ± ∞ \pm\infty ± ∞ f + ( x ) ≜ f ( x + ) f_+(x)\triangleq f(x+) f + ( x ) ≜ f ( x + ) f − ( x ) ≜ f ( x − ) f_-(x)\triangleq f(x-) f − ( x ) ≜ f ( x − ) f ( − ∞ ) ≜ lim x → − ∞ f ( x ) f(-\infty)\triangleq\lim_{x\to-\infty}f(x) f ( − ∞ ) ≜ lim x → − ∞ f ( x ) μ f \mu_f μ f R \mathbb{R} R x ∈ R x\in\mathbb{R} x ∈ R
μ f ( ( − ∞ , x ] ) = f + ( x ) − f ( − ∞ ) , μ f ( { x } ) = f + ( x ) − f − ( x ) , ∣ μ f ∣ ( R ) = TV ( f + ) ≤ TV ( f ) . (10) \mu_f((-\infty,x])=f_+(x)-f(-\infty), \qquad\mu_f(\{x\})=f_+(x)-f_-(x), \qquad|\mu_f|(\mathbb{R})=\operatorname{TV}(f_+)\leq\operatorname{TV}(f). \tag*{(10)} μ f (( − ∞ , x ]) = f + ( x ) − f ( − ∞ ) , μ f ({ x }) = f + ( x ) − f − ( x ) , ∣ μ f ∣ ( R ) = TV ( f + ) ≤ TV ( f ) . (10) see [48 Section 3.5] . Condition (J) holds for continuous functions and for right- or left-continuous functions; it excludes spurious values at removable discontinuities, such as those of 1 { 0 } 1_{\{0\}} 1 { 0 }
For fixed finite nonempty S ⊂ R S \subset\mathbb{R} S ⊂ R α ∈ ( 0 , 1 ] \alpha\in(0,1] α ∈ ( 0 , 1 ] K S α \mathcal{K}^{\alpha}_{S} K S α R ∖ S \mathbb{R} \setminus S R ∖ S α \alpha α S S S C 0 , α C^{0,\alpha} C 0 , α α = 1 \alpha= 1 α = 1 K S 1 \mathcal{K}^{1}_{S} K S 1 R \mathbb{R} R
Example 2.1 (Test observables). Fix a threshold v f > 0 v_f > 0 v f > 0
(a) The crossing indicator f i n d ( v ) ≜ 1 { ∣ v ∣ ≥ v f } f_{\mathrm{ind}}(v) \triangleq1_{\{|v|\ge v_f\}} f ind ( v ) ≜ 1 { ∣ v ∣ ≥ v f } B V J \mathrm{BV}_{\mathrm{J}} BV J T V ( f i n d ) = 2 \mathrm{TV}(f_{\mathrm{ind}}) = 2 TV ( f ind ) = 2
f i n d ∉ C 0 , α for every α ∈ ( 0 , 1 ] . f_{\mathrm{ind}} \notin C^{0,\alpha} \qquad\text{for every } \alpha\in(0,1]. f ind ∈ / C 0 , α for every α ∈ ( 0 , 1 ] . (b) The severity family
f α ( v ) ≜ min { ( ( ∣ v ∣ − v f ) + ) α , 1 } , α ∈ ( 0 , 1 ] , f_{\alpha}(v) \triangleq\min\{((|v|-v_f)_{+})^{\alpha},1\}, \qquad\alpha\in(0,1], f α ( v ) ≜ min {(( ∣ v ∣ − v f ) + ) α , 1 } , α ∈ ( 0 , 1 ] , satisfies 0 ≤ f α ≤ 1 0 \le f_{\alpha} \le1 0 ≤ f α ≤ 1 T V ( f α ) = 2 \mathrm{TV}(f_{\alpha}) = 2 TV ( f α ) = 2 [ f α ] α ≤ 1 [f_{\alpha}]_{\alpha} \le1 [ f α ] α ≤ 1
f α ∈ K { ± v f } α , [ f α ] α , { ± v f } = α . f_{\alpha} \in\mathcal{K}^{\alpha}_{\{\pm v_f\}}, \qquad[f_{\alpha}]_{\alpha,\{\pm v_f\}} = \alpha. f α ∈ K { ± v f } α , [ f α ] α , { ± v f } = α . For α = 1 \alpha= 1 α = 1 f α f_{\alpha} f α α \alpha α
f α → f i n d pointwise on { ∣ v ∣ ≠ v f } as α ↓ 0. f_{\alpha} \to f_{\mathrm{ind}} \quad\text{pointwise on } \{|v| \ne v_f\} \text{ as } \alpha\downarrow0. f α → f ind pointwise on { ∣ v ∣ = v f } as α ↓ 0. (c) The smooth anchor f s m ( v ) ≜ tanh ( ( v / v f ) 2 ) f_{\mathrm{sm}}(v) \triangleq\tanh((v/v_f)^2) f sm ( v ) ≜ tanh (( v / v f ) 2 ) T V ( f s m ) = 2 \mathrm{TV}(f_{\mathrm{sm}}) = 2 TV ( f sm ) = 2
Estimators
Let ( W i , η 0 i ) i ≥ 1 (\mathbf{W}^{i},\eta^{i}_{0})_{i\ge1} ( W i , η 0 i ) i ≥ 1 ( W , η 0 ) (\mathbf{W},\eta_{0}) ( W , η 0 ) ( V ε , i , V 0 , i ) (V^{\varepsilon,i},V^{0,i}) ( V ε , i , V 0 , i ) (9) ) driven by ( W i , η 0 i ) (\mathbf{W}^{i},\eta^{i}_{0}) ( W i , η 0 i ) F ∙ ≜ f ( V ∙ ( T ) ) F^{\bullet} \triangleq f(V^{\bullet}(T)) F ∙ ≜ f ( V ∙ ( T )) F ∙ , i ≜ f ( V ∙ , i ( T ) ) F^{\bullet,i} \triangleq f(V^{\bullet,i}(T)) F ∙ , i ≜ f ( V ∙ , i ( T )) ∙ ∈ { ε , 0 } \bullet\in\{\varepsilon,0\} ∙ ∈ { ε , 0 } I 0 ≜ E [ F 0 ] I^{0} \triangleq\mathbb{E}[F^{0}] I 0 ≜ E [ F 0 ] [27 ] , we consider the standard Monte Carlo, the control variate and the practical optimal control variate estimators
I ^ N ε ≜ 1 N ∑ i = 1 N F ε , i , J ^ N ε ≜ I 0 + 1 N ∑ i = 1 N ( F ε , i − F 0 , i ) , K ^ N ε ≜ λ ^ N ε I 0 + 1 N ∑ i = 1 N ( F ε , i − λ ^ N ε F 0 , i ) , (11) \widehat{I}^{\varepsilon}_{N} \triangleq\frac{1}{N}\sum_{i=1}^{N}F^{\varepsilon,i}, \qquad
\widehat{J}^{\varepsilon}_{N} \triangleq I^{0}+\frac{1}{N}\sum_{i=1}^{N}(F^{\varepsilon,i}-F^{0,i}), \qquad
\widehat{K}^{\varepsilon}_{N} \triangleq\widehat{\lambda}^{\varepsilon}_{N}I^{0}+\frac{1}{N}\sum_{i=1}^{N}(F^{\varepsilon,i}-\widehat{\lambda}^{\varepsilon}_{N}F^{0,i}), \tag*{(11)} I N ε ≜ N 1 i = 1 ∑ N F ε , i , J N ε ≜ I 0 + N 1 i = 1 ∑ N ( F ε , i − F 0 , i ) , K N ε ≜ λ N ε I 0 + N 1 i = 1 ∑ N ( F ε , i − λ N ε F 0 , i ) , (11) where λ ^ N ε \widehat{\lambda}^{\varepsilon}_{N} λ N ε F ε , i F^{\varepsilon,i} F ε , i F 0 , i F^{0,i} F 0 , i 0 0 0 F 0 , i F^{0,i} F 0 , i λ ε ≜ Cov ( F ε , F 0 ) / Var ( F 0 ) \lambda^{\varepsilon} \triangleq\operatorname{Cov}(F^{\varepsilon},F^{0})/\operatorname{Var}(F^{0}) λ ε ≜ Cov ( F ε , F 0 ) / Var ( F 0 ) λ ε ≜ 0 \lambda^{\varepsilon} \triangleq0 λ ε ≜ 0 Var ( F 0 ) = 0 \operatorname{Var}(F^{0})=0 Var ( F 0 ) = 0 F ε , F 0 F^{\varepsilon},F^{0} F ε , F 0
σ I ε 2 ≜ Var ( F ε ) , σ J ε 2 ≜ Var ( F ε − F 0 ) , σ K ε 2 ≜ min λ ∈ R Var ( F ε − λ F 0 ) , (12) \sigma^{2}_{I^{\varepsilon}} \triangleq\operatorname{Var}(F^{\varepsilon}), \qquad
\sigma^{2}_{J^{\varepsilon}} \triangleq\operatorname{Var}(F^{\varepsilon}-F^{0}), \qquad
\sigma^{2}_{K^{\varepsilon}} \triangleq\min_{\lambda\in\mathbb{R}}\operatorname{Var}(F^{\varepsilon}-\lambda F^{0}), \tag*{(12)} σ I ε 2 ≜ Var ( F ε ) , σ J ε 2 ≜ Var ( F ε − F 0 ) , σ K ε 2 ≜ λ ∈ R min Var ( F ε − λ F 0 ) , (12) the minimum being attained at λ = λ ε \lambda=\lambda^{\varepsilon} λ = λ ε σ K ε 2 ≤ min { σ I ε 2 , σ J ε 2 } \sigma^{2}_{K^{\varepsilon}} \le\min\{\sigma^{2}_{I^{\varepsilon}},\sigma^{2}_{J^{\varepsilon}}\} σ K ε 2 ≤ min { σ I ε 2 , σ J ε 2 } λ ∈ { 0 , 1 } \lambda\in\{0,1\} λ ∈ { 0 , 1 } ρ ε ≜ Cov ( F ε , F 0 ) / ( σ I ε Var ( F 0 ) 1 / 2 ) \rho^{\varepsilon} \triangleq\operatorname{Cov}(F^{\varepsilon},F^{0})/(\sigma_{I^{\varepsilon}}\operatorname{Var}(F^{0})^{1/2}) ρ ε ≜ Cov ( F ε , F 0 ) / ( σ I ε Var ( F 0 ) 1/2 ) ρ ε ≜ 0 \rho^{\varepsilon}\triangleq0 ρ ε ≜ 0 σ I ε Var ( F 0 ) 1 / 2 = 0 \sigma_{I^{\varepsilon}}\operatorname{Var}(F^{0})^{1/2}=0 σ I ε Var ( F 0 ) 1/2 = 0 [27 Proposition 2.1] , if f ∈ C 0 , α ∪ B V f \in C^{0,\alpha} \cup\mathrm{BV} f ∈ C 0 , α ∪ BV Var ( F 0 ) > 0 \operatorname{Var}(F^{0})>0 Var ( F 0 ) > 0 I ^ N ε \widehat{I}^{\varepsilon}_{N} I N ε J ^ N ε \widehat{J}^{\varepsilon}_{N} J N ε (12) , and
σ K ε 2 = ( 1 − ( ρ ε ) 2 ) σ I ε 2 = σ J ε 2 − Var ( F 0 ) ( 1 − λ ε ) 2 . (13) \sigma^{2}_{K^{\varepsilon}}=(1-(\rho^{\varepsilon})^{2})\sigma^{2}_{I^{\varepsilon}}=\sigma^{2}_{J^{\varepsilon}}-\operatorname{Var}(F^{0})(1-\lambda^{\varepsilon})^{2}. \tag*{(13)} σ K ε 2 = ( 1 − ( ρ ε ) 2 ) σ I ε 2 = σ J ε 2 − Var ( F 0 ) ( 1 − λ ε ) 2 . (13) The control variates target I ε I^{\varepsilon} I ε I ε − I 0 I^{\varepsilon}-I^{0} I ε − I 0 I 0 = ∫ R f p T I^{0}=\int_{\mathbb{R}} f\,p_{T} I 0 = ∫ R f p T p T p_{T} p T V 0 ( T ) V^{0}(T) V 0 ( T ) Proposition B.3 ), known in closed form in regime (I) [10 , 11 , 12 ] and, in regimes (II)–(III), as a spectral series [12 ] and through its Laplace transform [13 ] ; thus I 0 I^{0} I 0 § 5.3 ).
Main results
Throughout this section, Assumptions A1, A2, and A3 are assumed to hold. For an observable f f f D ε ≜ f ( V ε ( T ) ) − f ( V 0 ( T ) ) D^{\varepsilon} \triangleq f(V^{\varepsilon}(T)) - f(V^{0}(T)) D ε ≜ f ( V ε ( T )) − f ( V 0 ( T )) Z ε ≜ Z ε ( T ) = V ε ( T ) − V 0 ( T ) Z^{\varepsilon} \triangleq Z^{\varepsilon}(T) = V^{\varepsilon}(T) - V^{0}(T) Z ε ≜ Z ε ( T ) = V ε ( T ) − V 0 ( T ) ϑ ε ≜ ε ( 1 + log ( 1 / ε ) ) 1 / 2 \vartheta_{\varepsilon} \triangleq\varepsilon(1+\log(1/\varepsilon))^{1/2} ϑ ε ≜ ε ( 1 + log ( 1/ ε ) ) 1/2 ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ] [27 Lemma 6.1] : there are C 1 > 0 C_1 > 0 C 1 > 0 ε G M ∈ ( 0 , T ] \varepsilon_{\mathrm{GM}} \in(0,T] ε GM ∈ ( 0 , T ]
E [ ∣ Z ε ∣ 2 ] ≤ C 1 ε 2 , ε ∈ ( 0 , ε G M ] , (14) \mathbb{E}\left[|Z^{\varepsilon}|^2\right] \le C_1\varepsilon^2,\qquad\varepsilon\in(0,\varepsilon_{\mathrm{GM}}], \tag*{(14)} E [ ∣ Z ε ∣ 2 ] ≤ C 1 ε 2 , ε ∈ ( 0 , ε GM ] , (14) and we set ε 0 ≜ ε G M ∧ 1 \varepsilon_0 \triangleq\varepsilon_{\mathrm{GM}} \wedge1 ε 0 ≜ ε GM ∧ 1 C C C Δ \Delta Δ γ \gamma γ ℓ \ell ℓ σ \sigma σ A \mathbf{A} A K \mathbf{K} K r r r d d d d ′ d' d ′ T T T v 0 v_0 v 0 ε \varepsilon ε
Variance rates
The proof of the following theorem is given in Section 4.1 4.1 4.1
Theorem 3.1. Let f f f ε > 0 \varepsilon> 0 ε > 0
σ K ε 2 ≤ σ J ε 2 ≤ E [ ( D ε ) 2 ] , (15) \sigma_{K_\varepsilon}^2 \le\sigma_{J_\varepsilon}^2 \le\mathbb{E}\left[(D^\varepsilon)^2\right], \tag*{(15)} σ K ε 2 ≤ σ J ε 2 ≤ E [ ( D ε ) 2 ] , (15) and, for all ε ∈ ( 0 , ε 0 ] \varepsilon\in(0,\varepsilon_0] ε ∈ ( 0 , ε 0 ] E [ ( D ε ) 2 ] \mathbb{E}[(D^\varepsilon)^2] E [( D ε ) 2 ] f f f
(a) if f ∈ C 0 , α f \in C^{0,\alpha} f ∈ C 0 , α α ∈ ( 0 , 1 ] \alpha\in(0,1] α ∈ ( 0 , 1 ] E [ ( D ε ) 2 ] ≤ [ f ] α 2 C 1 α / 2 ε 2 α \mathbb{E}[(D^\varepsilon)^2] \le[f]_\alpha^2 C_1^{\alpha/2}\varepsilon^{2\alpha} E [( D ε ) 2 ] ≤ [ f ] α 2 C 1 α /2 ε 2 α
(b) if f ∈ B V J f \in\mathrm{BV}_J f ∈ BV J E [ ( D ε ) 2 ] ≤ C TV ( f ) 2 ϑ ε \mathbb{E}[(D^\varepsilon)^2] \le C\operatorname{TV}(f)^2\vartheta_\varepsilon E [( D ε ) 2 ] ≤ C TV ( f ) 2 ϑ ε
(c) if f ∈ C 0 , α ∩ B V J f \in C^{0,\alpha} \cap\mathrm{BV}_J f ∈ C 0 , α ∩ BV J E [ ( D ε ) 2 ] ≤ C [ f ] α TV ( f ) ϑ ε 1 + α \mathbb{E}[(D^\varepsilon)^2] \le C[f]_\alpha\operatorname{TV}(f)\vartheta_\varepsilon^{1+\alpha} E [( D ε ) 2 ] ≤ C [ f ] α TV ( f ) ϑ ε 1 + α
(d) if f ∈ K S α f \in K_S^\alpha f ∈ K S α S S S α ∈ ( 0 , 1 ] \alpha\in(0,1] α ∈ ( 0 , 1 ]
E [ ( D ε ) 2 ] ≤ C α , S [ f ] α , S 2 ψ α ( ε ) , ψ α ( ε ) ≜ { ϑ ε 1 + 2 α , α < 1 / 2 , ϑ ε 2 ( 1 + log ( 1 / ε ) ) , α = 1 / 2 , ϑ ε 2 , α > 1 / 2. \mathbb{E}[(D^\varepsilon)^2] \le C_{\alpha,S}[f]_{\alpha,S}^2\psi_\alpha(\varepsilon),\qquad
\psi_\alpha(\varepsilon) \triangleq
\begin{cases}
\vartheta_\varepsilon^{1+2\alpha}, & \alpha< 1/2,\\
\vartheta_\varepsilon^2(1+\log(1/\varepsilon)), & \alpha= 1/2,\\
\vartheta_\varepsilon^2, & \alpha> 1/2.
\end{cases} E [( D ε ) 2 ] ≤ C α , S [ f ] α , S 2 ψ α ( ε ) , ψ α ( ε ) ≜ ⎩ ⎨ ⎧ ϑ ε 1 + 2 α , ϑ ε 2 ( 1 + log ( 1/ ε )) , ϑ ε 2 , α < 1/2 , α = 1/2 , α > 1/2. The constants C C C C α , S C_{\alpha,S} C α , S ε \varepsilon ε f f f α \alpha α C α , S C_{\alpha,S} C α , S α \alpha α ∣ S ∣ |S| ∣ S ∣ ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ]
Consequences
The proofs of the following corollaries are given in Section 4.2 4.2 4.2
Corollary 3.1 (Bias of the white-noise approximation). Always ∣ I ε − I 0 ∣ ≤ E [ ∣ D ε ∣ ] |I^\varepsilon-I^0| \le\mathbb{E}[|D^\varepsilon|] ∣ I ε − I 0 ∣ ≤ E [ ∣ D ε ∣ ]
(a) if f ∈ C 0 , α f \in C^{0,\alpha} f ∈ C 0 , α ∣ I ε − I 0 ∣ ≤ [ f ] α C 1 α / 2 ε α |I^\varepsilon-I^0| \le[f]_\alpha C_1^{\alpha/2}\varepsilon^\alpha ∣ I ε − I 0 ∣ ≤ [ f ] α C 1 α /2 ε α ε ∈ ( 0 , ε 0 ] \varepsilon\in(0,\varepsilon_0] ε ∈ ( 0 , ε 0 ]
(b) if f ∈ B V J f \in\mathrm{BV}_J f ∈ BV J ∣ I ε − I 0 ∣ ≤ C TV ( f ) ϑ ε |I^\varepsilon-I^0| \le C\operatorname{TV}(f)\vartheta_\varepsilon ∣ I ε − I 0 ∣ ≤ C TV ( f ) ϑ ε ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ]
(c) if f ∈ K S α f \in K_S^\alpha f ∈ K S α ∣ I ε − I 0 ∣ ≤ C α , S ′ [ f ] α , S ϑ ε |I^\varepsilon-I^0| \le C'_{\alpha,S}[f]_{\alpha,S}\vartheta_\varepsilon ∣ I ε − I 0 ∣ ≤ C α , S ′ [ f ] α , S ϑ ε ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ] C α , S ′ C'_{\alpha,S} C α , S ′ α \alpha α ∣ S ∣ |S| ∣ S ∣
Part (b) improves on the O ( ϑ ε 1 / 2 ) O(\vartheta_\varepsilon^{1/2}) O ( ϑ ε 1/2 ) ∣ D ε ∣ |D^\varepsilon| ∣ D ε ∣ O ( ε 2 ) O(\varepsilon^2) O ( ε 2 ) [27 Lemma 6.1] .
Corollary 3.2 (Variance reduction). Let f f f Var ( F 0 ) > 0 \operatorname{Var}(F^0)>0 Var ( F 0 ) > 0 R f ( ε ) R_f(\varepsilon) R f ( ε ) E [ ( D ε ) 2 ] \mathbb{E}[(D^\varepsilon)^2] E [( D ε ) 2 ] σ 0 ≜ Var ( F 0 ) 1 / 2 \sigma_0 \triangleq\operatorname{Var}(F^0)^{1/2} σ 0 ≜ Var ( F 0 ) 1/2 ∣ σ I ε − σ 0 ∣ ≤ σ J ε |\sigma_{I_\varepsilon}-\sigma_0| \le\sigma_{J_\varepsilon} ∣ σ I ε − σ 0 ∣ ≤ σ J ε σ I ε → σ 0 \sigma_{I_\varepsilon}\to\sigma_0 σ I ε → σ 0 ε ∈ ( 0 , ε 0 ] \varepsilon\in(0,\varepsilon_0] ε ∈ ( 0 , ε 0 ] R f ( ε ) ≤ σ 0 2 / 4 R_f(\varepsilon)\le\sigma_0^2/4 R f ( ε ) ≤ σ 0 2 /4
σ K ε 2 σ I ε 2 = 1 − ( ρ ε ) 2 ≤ σ J ε 2 σ 0 2 ≤ 4 R f ( ε ) σ 0 2 , ∣ 1 − λ ε ∣ ≤ σ J ε σ 0 ≤ R f ( ε ) 1 / 2 σ 0 . \frac{\sigma_{K_\varepsilon}^2}{\sigma_{I_\varepsilon}^2}
=1-(\rho^\varepsilon)^2
\le\frac{\sigma_{J_\varepsilon}^2}{\sigma_0^2}
\le\frac{4R_f(\varepsilon)}{\sigma_0^2},
\qquad
|1-\lambda^\varepsilon|
\le\frac{\sigma_{J_\varepsilon}}{\sigma_0}
\le\frac{R_f(\varepsilon)^{1/2}}{\sigma_0}. σ I ε 2 σ K ε 2 = 1 − ( ρ ε ) 2 ≤ σ 0 2 σ J ε 2 ≤ σ 0 2 4 R f ( ε ) , ∣1 − λ ε ∣ ≤ σ 0 σ J ε ≤ σ 0 R f ( ε ) 1/2 . Since a coupled path costs about twice a colored-noise path and I 0 I^{0} I 0 J ^ N ε \widehat{J}^{\varepsilon}_{N} J N ε 2 σ J ε 2 / σ I ε 2 = O ( R f ( ε ) ) 2\sigma_{J^{\varepsilon}}^{2}/\sigma_{I^{\varepsilon}}^{2}=O(\mathcal{R}_{f}(\varepsilon)) 2 σ J ε 2 / σ I ε 2 = O ( R f ( ε )) [27 Section 8] when I 0 I^{0} I 0
Corollary 3.3 (Non-asymptotic sample sizes). Let f f f osc ( f ) ≜ sup f − inf f \operatorname{osc}(f)\triangleq\sup f-\inf f osc ( f ) ≜ sup f − inf f t > 0 t>0 t > 0 ϱ ∈ ( 0 , 1 ) \varrho\in(0,1) ϱ ∈ ( 0 , 1 )
P ( ∣ J ^ N ε − I ε ∣ ≥ t ) ≤ 2 exp ( − N t 2 2 σ J ε 2 + 4 3 osc ( f ) t ) . (16) \mathbb{P}\left(\left|\widehat{J}^{\varepsilon}_{N}-I^{\varepsilon}\right|\geq t\right)\leq2\exp\left(-\frac{Nt^{2}}{2\sigma_{J^{\varepsilon}}^{2}+\frac{4}{3}\operatorname{osc}(f)t}\right). \tag*{(16)} P ( J N ε − I ε ≥ t ) ≤ 2 exp ( − 2 σ J ε 2 + 3 4 osc ( f ) t N t 2 ) . (16) Hence ∣ J ^ N ε − I ε ∣ < t \left|\widehat{J}^{\varepsilon}_{N}-I^{\varepsilon}\right|<t J N ε − I ε < t 1 − ϱ 1-\varrho 1 − ϱ
N ≥ ( 2 σ J ε 2 t 2 + 4 osc ( f ) 3 t ) log 2 ϱ , against N ≥ ( 2 σ I ε 2 t 2 + 2 osc ( f ) 3 t ) log 2 ϱ (17) N\geq\left(\frac{2\sigma_{J^{\varepsilon}}^{2}}{t^{2}}+\frac{4\operatorname{osc}(f)}{3t}\right)\log\frac{2}{\varrho},
\qquad\text{against}\qquad
N\geq\left(\frac{2\sigma_{I^{\varepsilon}}^{2}}{t^{2}}+\frac{2\operatorname{osc}(f)}{3t}\right)\log\frac{2}{\varrho} \tag*{(17)} N ≥ ( t 2 2 σ J ε 2 + 3 t 4 osc ( f ) ) log ϱ 2 , against N ≥ ( t 2 2 σ I ε 2 + 3 t 2 osc ( f ) ) log ϱ 2 (17) for I ^ N ε \widehat{I}^{\varepsilon}_{N} I N ε 2 σ J ε 2 t − 2 2\sigma_{J^{\varepsilon}}^{2}t^{-2} 2 σ J ε 2 t − 2 2 R f ( ε ) t − 2 2\mathcal{R}_{f}(\varepsilon)t^{-2} 2 R f ( ε ) t − 2 R f \mathcal{R}_{f} R f 2 Var ( F 0 ) t − 2 2\operatorname{Var}(F^{0})t^{-2} 2 Var ( F 0 ) t − 2
Proofs of the main results
In this section, Assumptions A1, A2, and A3 are assumed to hold, and D ε D^{\varepsilon} D ε Z ε Z^{\varepsilon} Z ε ϑ ε \vartheta_{\varepsilon} ϑ ε ε 0 \varepsilon_{0} ε 0 § 3 . Besides (14) , we use the bound M M M V 0 ( T ) V^{0}(T) V 0 ( T ) ϑ ε \vartheta_{\varepsilon} ϑ ε u ∈ R u\in\mathbb{R} u ∈ R
E u ≜ { u ∈ [ [ V 0 ( T ) , V ε ( T ) ] ] } ⊂ { ∣ V 0 ( T ) − u ∣ ≤ ∣ Z ε ∣ } ; E_{u}\triangleq\left\{u\in\left[\!\left[V^{0}(T),V^{\varepsilon}(T)\right]\!\right]\right\}\subset\left\{\left|V^{0}(T)-u\right|\leq\left|Z^{\varepsilon}\right|\right\}; E u ≜ { u ∈ [ [ V 0 ( T ) , V ε ( T ) ] ] } ⊂ { V 0 ( T ) − u ≤ ∣ Z ε ∣ } ; the map ( ω , u ) ↦ 1 E u ( ω ) (\omega,u)\mapsto\mathbf{1}_{E_{u}}(\omega) ( ω , u ) ↦ 1 E u ( ω ) F ⊗ B ( R ) \mathcal{F}\otimes\mathcal{B}(\mathbb{R}) F ⊗ B ( R ) E u = { V 0 ( T ) ∧ V ε ( T ) ≤ u ≤ V 0 ( T ) ∨ V ε ( T ) } E_{u}=\{V^{0}(T)\wedge V^{\varepsilon}(T)\leq u\leq V^{0}(T)\vee V^{\varepsilon}(T)\} E u = { V 0 ( T ) ∧ V ε ( T ) ≤ u ≤ V 0 ( T ) ∨ V ε ( T )} θ > 0 \theta>0 θ > 0 E u ⊂ { ∣ Z ε ∣ > θ } ∪ { ∣ V 0 ( T ) − u ∣ ≤ θ } E_{u}\subset\{|Z^{\varepsilon}|>\theta\}\cup\{|V^{0}(T)-u|\leq\theta\} E u ⊂ { ∣ Z ε ∣ > θ } ∪ { ∣ V 0 ( T ) − u ∣ ≤ θ }
P ( E u ) ≤ P ( ∣ Z ε ∣ > θ ) + 2 M θ , u ∈ R , θ > 0. (18) \mathbb{P}(E_{u})\leq\mathbb{P}\left(|Z^{\varepsilon}|>\theta\right)+2M\theta,\qquad u\in\mathbb{R},\ \theta>0. \tag*{(18)} P ( E u ) ≤ P ( ∣ Z ε ∣ > θ ) + 2 M θ , u ∈ R , θ > 0. (18) If f ∈ B V J f\in\mathrm{BV}_{J} f ∈ BV J ∣ f ( x ) − f ( y ) ∣ ≤ ∣ μ f ∣ ( [ [ x , y ] ] ) |f(x)-f(y)|\leq|\mu_{f}|([[x,y]]) ∣ f ( x ) − f ( y ) ∣ ≤ ∣ μ f ∣ ([[ x , y ]])
∣ D ε ∣ ≤ ∫ R 1 E u ∣ μ f ∣ ( d u ) , ( D ε ) 2 ≤ TV ( f ) ∫ R 1 E u ∣ μ f ∣ ( d u ) , (19) |D^{\varepsilon}|\leq\int_{\mathbb{R}}\mathbf{1}_{E_{u}}|\mu_{f}|(\mathrm{d}u),\qquad
(D^{\varepsilon})^{2}\leq\operatorname{TV}(f)\int_{\mathbb{R}}\mathbf{1}_{E_{u}}|\mu_{f}|(\mathrm{d}u), \tag*{(19)} ∣ D ε ∣ ≤ ∫ R 1 E u ∣ μ f ∣ ( d u ) , ( D ε ) 2 ≤ TV ( f ) ∫ R 1 E u ∣ μ f ∣ ( d u ) , (19) and, by Tonelli’s theorem and ∣ μ f ∣ ( R ) ≤ TV ( f ) |\mu_{f}|(\mathbb{R})\leq\operatorname{TV}(f) ∣ μ f ∣ ( R ) ≤ TV ( f )
E [ ∣ D ε ∣ ] ≤ TV ( f ) sup u ∈ R P ( E u ) , E [ ( D ε ) 2 ] ≤ TV ( f ) 2 sup u ∈ R P ( E u ) . (20) \mathbb{E}[|D^{\varepsilon}|]\leq\operatorname{TV}(f)\sup_{u\in\mathbb{R}}\mathbb{P}(E_{u}),\qquad
\mathbb{E}[(D^{\varepsilon})^{2}]\leq\operatorname{TV}(f)^{2}\sup_{u\in\mathbb{R}}\mathbb{P}(E_{u}). \tag*{(20)} E [ ∣ D ε ∣ ] ≤ TV ( f ) u ∈ R sup P ( E u ) , E [( D ε ) 2 ] ≤ TV ( f ) 2 u ∈ R sup P ( E u ) . (20) Proof of Theorem 3.1
The inequalities (15) follow from (12) , with λ = 1 \lambda=1 λ = 1 Var ( D ε ) ≤ E [ ( D ε ) 2 ] \operatorname{Var}(D^{\varepsilon})\leq\mathbb{E}[(D^{\varepsilon})^{2}] Var ( D ε ) ≤ E [( D ε ) 2 ] ε ↦ ϑ ε \varepsilon\mapsto\vartheta_{\varepsilon} ε ↦ ϑ ε ( 0 , 1 ] (0,1] ( 0 , 1 ] ϑ 1 = 1 \vartheta_{1}=1 ϑ 1 = 1 ε ≤ ϑ ε ≤ 1 \varepsilon\leq\vartheta_{\varepsilon}\leq1 ε ≤ ϑ ε ≤ 1 ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ] ε \varepsilon ε p , q > 0 p,q>0 p , q > 0 ∣ Z ε ∣ ≤ sup t ≤ T ∣ Z ε ( t ) ∣ |Z^{\varepsilon}|\leq\sup_{t\leq T}|Z^{\varepsilon}(t)| ∣ Z ε ∣ ≤ sup t ≤ T ∣ Z ε ( t ) ∣
E [ ∣ Z ε ∣ p ] ≤ C p ϑ ε p , P ( ∣ Z ε ∣ > K q ϑ ε ) ≤ ( T + 1 ) C 0 ε q . (21) \mathbb{E}[|Z^{\varepsilon}|^{p}]\leq C_{p}\vartheta_{\varepsilon}^{p},\qquad
\mathbb{P}(|Z^{\varepsilon}|>K_{q}\vartheta_{\varepsilon})\leq(T+1)C_{0}\varepsilon^{q}. \tag*{(21)} E [ ∣ Z ε ∣ p ] ≤ C p ϑ ε p , P ( ∣ Z ε ∣ > K q ϑ ε ) ≤ ( T + 1 ) C 0 ε q . (21) (a) Since ∣ D ε ∣ ≤ [ f ] α ∣ Z ε ∣ α |D^{\varepsilon}|\leq[f]_{\alpha}|Z^{\varepsilon}|^{\alpha} ∣ D ε ∣ ≤ [ f ] α ∣ Z ε ∣ α u ↦ u α u\mapsto u^{\alpha} u ↦ u α [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) (14) give, for ε ∈ ( 0 , ε G M ] \varepsilon\in(0,\varepsilon_{\mathrm{GM}}] ε ∈ ( 0 , ε GM ] E [ ( D ε ) 2 ] ≤ [ f ] α 2 ( E [ ∣ Z ε ∣ 2 ] ) α ≤ [ f ] α 2 C 1 2 α ε 2 α \mathbb{E}[(D^{\varepsilon})^{2}]\leq[f]_{\alpha}^{2}(\mathbb{E}[|Z^{\varepsilon}|^{2}])^{\alpha}\leq[f]_{\alpha}^{2}C_{1}^{2\alpha}\varepsilon^{2\alpha} E [( D ε ) 2 ] ≤ [ f ] α 2 ( E [ ∣ Z ε ∣ 2 ] ) α ≤ [ f ] α 2 C 1 2 α ε 2 α
(b) Take θ = K 1 ϑ ε \theta=K_{1}\vartheta_{\varepsilon} θ = K 1 ϑ ε (18) . By (21) , P ( E u ) ≤ ( T + 1 ) C 0 ε + 2 M K 1 ϑ ε ≤ ( ( T + 1 ) C 0 + 2 M K 1 ) ϑ ε \mathbb{P}(E_{u})\leq(T+1)C_{0}\varepsilon+2MK_{1}\vartheta_{\varepsilon}\leq((T+1)C_{0}+2MK_{1})\vartheta_{\varepsilon} P ( E u ) ≤ ( T + 1 ) C 0 ε + 2 M K 1 ϑ ε ≤ (( T + 1 ) C 0 + 2 M K 1 ) ϑ ε u u u (20) gives the claim.
(c) Combining ∣ D ε ∣ ≤ [ f ] α ∣ Z ε ∣ α |D^{\varepsilon}|\leq[f]_{\alpha}|Z^{\varepsilon}|^{\alpha} ∣ D ε ∣ ≤ [ f ] α ∣ Z ε ∣ α (19) , ( D ε ) 2 = ∣ D ε ∣ ∣ D ε ∣ ≤ [ f ] α ∣ Z ε ∣ α ∫ R 1 E u ∣ μ f ∣ ( d u ) (D^{\varepsilon})^{2}=|D^{\varepsilon}||D^{\varepsilon}|\leq[f]_{\alpha}|Z^{\varepsilon}|^{\alpha}\int_{\mathbb{R}}\mathbf{1}_{E_{u}}|\mu_{f}|(\mathrm{d}u) ( D ε ) 2 = ∣ D ε ∣∣ D ε ∣ ≤ [ f ] α ∣ Z ε ∣ α ∫ R 1 E u ∣ μ f ∣ ( d u ) E [ ( D ε ) 2 ] ≤ [ f ] α TV ( f ) sup u E [ ∣ Z ε ∣ α 1 E u ] \mathbb{E}[(D^{\varepsilon})^{2}]\leq[f]_{\alpha}\operatorname{TV}(f)\sup_{u}\mathbb{E}[|Z^{\varepsilon}|^{\alpha}\mathbf{1}_{E_{u}}] E [( D ε ) 2 ] ≤ [ f ] α TV ( f ) sup u E [ ∣ Z ε ∣ α 1 E u ] θ = K 2 ϑ ε \theta=K_{2}\vartheta_{\varepsilon} θ = K 2 ϑ ε E u ∩ { ∣ Z ε ∣ ≤ θ } E_u \cap\{|Z^\varepsilon| \leq\theta\} E u ∩ { ∣ Z ε ∣ ≤ θ } ∣ V 0 ( T ) − u ∣ ≤ θ |V^0(T)-u| \leq\theta ∣ V 0 ( T ) − u ∣ ≤ θ { ∣ Z ε ∣ > θ } \{|Z^\varepsilon| > \theta\} { ∣ Z ε ∣ > θ }
E [ ∣ Z ε ∣ α 1 E u ] ≤ θ α P ( ∣ V 0 ( T ) − u ∣ ≤ θ ) + ( E [ ∣ Z ε ∣ 2 α ] ) 1 / 2 P ( ∣ Z ε ∣ > θ ) 1 / 2 \mathbb{E}\left[|Z^\varepsilon|^\alpha1_{E_u}\right] \leq\theta^\alpha\mathbb{P}\left(|V^0(T)-u| \leq\theta\right) + \left(\mathbb{E}\left[|Z^\varepsilon|^{2\alpha}\right]\right)^{1/2}\mathbb{P}\left(|Z^\varepsilon| > \theta\right)^{1/2} E [ ∣ Z ε ∣ α 1 E u ] ≤ θ α P ( ∣ V 0 ( T ) − u ∣ ≤ θ ) + ( E [ ∣ Z ε ∣ 2 α ] ) 1/2 P ( ∣ Z ε ∣ > θ ) 1/2 ≤ 2 M K 1 2 + α ϑ ε 1 + α + C 2 1 / 2 ϑ ε α ( ( T + 1 ) C 0 ) 1 / 2 ε , \leq2M K_1^{2+\alpha}\vartheta_\varepsilon^{1+\alpha} + C_2^{1/2}\vartheta_\varepsilon^{\alpha}((T+1)C_0)^{1/2}\varepsilon, ≤ 2 M K 1 2 + α ϑ ε 1 + α + C 2 1/2 ϑ ε α (( T + 1 ) C 0 ) 1/2 ε , by (21) with q = 2 q=2 q = 2 C ϑ ε 1 + α C\vartheta_\varepsilon^{1+\alpha} C ϑ ε 1 + α ε ≤ ϑ ε \varepsilon\leq\vartheta_\varepsilon ε ≤ ϑ ε
(d) Let f ∈ K S α f \in\mathcal{K}^{\alpha}_{S} f ∈ K S α θ ≜ K 4 ϑ ε \theta\triangleq K_4\vartheta_\varepsilon θ ≜ K 4 ϑ ε Υ ≜ dist ( V 0 ( T ) , S ) \Upsilon\triangleq\operatorname{dist}(V^0(T),S) Υ ≜ dist ( V 0 ( T ) , S ) Proposition B.3 , for x ≥ 0 x \geq0 x ≥ 0 h > 0 h > 0 h > 0
P ( Υ ∈ [ x , x + h ] ) ≤ ∑ ς ∈ S P ( ∣ V 0 ( T ) − ς ∣ ∈ [ x , x + h ] ) ≤ 2 ∣ S ∣ M h , \mathbb{P}(\Upsilon\in[x,x+h]) \leq\sum_{\varsigma\in S}\mathbb{P}(|V^0(T)-\varsigma| \in[x,x+h]) \leq2|S|Mh, P ( Υ ∈ [ x , x + h ]) ≤ ς ∈ S ∑ P ( ∣ V 0 ( T ) − ς ∣ ∈ [ x , x + h ]) ≤ 2∣ S ∣ M h , i.e., the law of Υ \Upsilon Υ 2 ∣ S ∣ M 2|S|M 2∣ S ∣ M E [ ( D ε ) 2 ] = ( i ) + ( i i ) + ( i i i ) \mathbb{E}[(D^\varepsilon)^2]=(i)+(ii)+(iii) E [( D ε ) 2 ] = ( i ) + ( ii ) + ( iii ) { ∣ Z ε ∣ > θ } \{|Z^\varepsilon|>\theta\} { ∣ Z ε ∣ > θ } { ∣ Z ε ∣ ≤ θ , Υ ≤ 2 θ } \{|Z^\varepsilon|\leq\theta,\Upsilon\leq2\theta\} { ∣ Z ε ∣ ≤ θ , Υ ≤ 2 θ } { ∣ Z ε ∣ ≤ θ , Υ > 2 θ } \{|Z^\varepsilon|\leq\theta,\Upsilon>2\theta\} { ∣ Z ε ∣ ≤ θ , Υ > 2 θ } ∣ D ε ∣ ≤ c α , S [ f ] α , S ∣ Z ε ∣ α |D^\varepsilon|\leq c_{\alpha,S}[f]_{\alpha,S}|Z^\varepsilon|^\alpha ∣ D ε ∣ ≤ c α , S [ f ] α , S ∣ Z ε ∣ α (21) with q = 4 q=4 q = 4
( i ) ≤ c α , S 2 [ f ] α , S 2 ( E [ ∣ Z ε ∣ 4 α ] ) 1 / 2 P ( ∣ Z ε ∣ > θ ) 1 / 2 ≤ C c α , S 2 [ f ] α , S 2 ϑ ε 2 α ε 2 ≤ C c α , S 2 [ f ] α , S 2 ϑ ε 2 + 2 α , (i)\leq c_{\alpha,S}^2[f]_{\alpha,S}^2\left(\mathbb{E}[|Z^\varepsilon|^{4\alpha}]\right)^{1/2}\mathbb{P}(|Z^\varepsilon|>\theta)^{1/2}\leq Cc_{\alpha,S}^2[f]_{\alpha,S}^2\vartheta_\varepsilon^{2\alpha}\varepsilon^2\leq Cc_{\alpha,S}^2[f]_{\alpha,S}^2\vartheta_\varepsilon^{2+2\alpha}, ( i ) ≤ c α , S 2 [ f ] α , S 2 ( E [ ∣ Z ε ∣ 4 α ] ) 1/2 P ( ∣ Z ε ∣ > θ ) 1/2 ≤ C c α , S 2 [ f ] α , S 2 ϑ ε 2 α ε 2 ≤ C c α , S 2 [ f ] α , S 2 ϑ ε 2 + 2 α , and
( i i ) ≤ c α , S 2 [ f ] α , S 2 θ 2 α P ( Υ ≤ 2 θ ) ≤ 4 ∣ S ∣ M c α , S 2 [ f ] α , S 2 θ 1 + 2 α . (ii)\leq c_{\alpha,S}^2[f]_{\alpha,S}^2\theta^{2\alpha}\mathbb{P}(\Upsilon\leq2\theta)\leq4|S|M c_{\alpha,S}^2[f]_{\alpha,S}^2\theta^{1+2\alpha}. ( ii ) ≤ c α , S 2 [ f ] α , S 2 θ 2 α P ( Υ ≤ 2 θ ) ≤ 4∣ S ∣ M c α , S 2 [ f ] α , S 2 θ 1 + 2 α . On the event of (iii), every point u u u [ [ V 0 ( T ) , V ε ( T ) ] ] [[V^0(T),V^\varepsilon(T)]] [[ V 0 ( T ) , V ε ( T )]] dist ( u , S ) ≥ Υ − ∣ Z ε ∣ ≥ Υ / 2 \operatorname{dist}(u,S)\geq\Upsilon-|Z^\varepsilon|\geq\Upsilon/2 dist ( u , S ) ≥ Υ − ∣ Z ε ∣ ≥ Υ/2 ∣ D ε ∣ ≤ [ f ] α , S ( Υ / 2 ) α − 1 ∣ Z ε ∣ ≤ 2 [ f ] α , S Υ α − 1 θ |D^\varepsilon|\leq[f]_{\alpha,S}(\Upsilon/2)^{\alpha-1}|Z^\varepsilon|\leq2[f]_{\alpha,S}\Upsilon^{\alpha-1}\theta ∣ D ε ∣ ≤ [ f ] α , S ( Υ/2 ) α − 1 ∣ Z ε ∣ ≤ 2 [ f ] α , S Υ α − 1 θ Υ 2 α − 2 ≤ 1 \Upsilon^{2\alpha-2}\leq1 Υ 2 α − 2 ≤ 1 Υ ≥ 1 \Upsilon\geq1 Υ ≥ 1
( i i i ) ≤ 4 [ f ] α , S 2 θ 2 E [ Υ 2 α − 2 1 { Υ > 2 θ } ] ≤ 4 [ f ] α , S 2 θ 2 ( 1 + 2 ∣ S ∣ M ∫ 2 θ ∧ 1 1 x 2 α − 2 d x ) . (iii)\leq4[f]_{\alpha,S}^2\theta^2\mathbb{E}\left[\Upsilon^{2\alpha-2}1_{\{\Upsilon>2\theta\}}\right]\leq4[f]_{\alpha,S}^2\theta^2\left(1+2|S|M\int_{2\theta\wedge1}^{1}x^{2\alpha-2}\,dx\right). ( iii ) ≤ 4 [ f ] α , S 2 θ 2 E [ Υ 2 α − 2 1 { Υ > 2 θ } ] ≤ 4 [ f ] α , S 2 θ 2 ( 1 + 2∣ S ∣ M ∫ 2 θ ∧ 1 1 x 2 α − 2 d x ) . The last integral vanishes if 2 θ ≥ 1 2\theta\geq1 2 θ ≥ 1 ( 2 θ ) 2 α − 1 / ( 1 − 2 α ) (2\theta)^{2\alpha-1}/(1-2\alpha) ( 2 θ ) 2 α − 1 / ( 1 − 2 α ) α < 1 / 2 \alpha<1/2 α < 1/2 log ( 1 / ( 2 θ ) ) \log(1/(2\theta)) log ( 1/ ( 2 θ )) α = 1 / 2 \alpha=1/2 α = 1/2 1 / ( 2 α − 1 ) 1/(2\alpha-1) 1/ ( 2 α − 1 ) α > 1 / 2 \alpha>1/2 α > 1/2 θ = K 4 ϑ ε \theta=K_4\vartheta_\varepsilon θ = K 4 ϑ ε ε ≤ ϑ ε ≤ 1 \varepsilon\leq\vartheta_\varepsilon\leq1 ε ≤ ϑ ε ≤ 1 log ( 1 / ( 2 θ ) ) ≤ log + ( 1 / ( 2 K 4 ) ) + log ( 1 / ε ) \log(1/(2\theta))\leq\log_+(1/(2K_4))+\log(1/\varepsilon) log ( 1/ ( 2 θ )) ≤ log + ( 1/ ( 2 K 4 )) + log ( 1/ ε ) E [ ( D ε ) 2 ] ≤ C α , S [ f ] α , S 2 ψ α ( ε ) \mathbb{E}[(D^\varepsilon)^2]\leq C_{\alpha,S}[f]_{\alpha,S}^2\psi_\alpha(\varepsilon) E [( D ε ) 2 ] ≤ C α , S [ f ] α , S 2 ψ α ( ε ) α < 1 / 2 \alpha<1/2 α < 1/2 θ 2 ( 2 θ ) 2 α − 1 ≤ θ 1 + 2 α \theta^2(2\theta)^{2\alpha-1}\leq\theta^{1+2\alpha} θ 2 ( 2 θ ) 2 α − 1 ≤ θ 1 + 2 α ϑ ε 2 ≤ ϑ ε 1 + 2 α \vartheta_\varepsilon^2\leq\vartheta_\varepsilon^{1+2\alpha} ϑ ε 2 ≤ ϑ ε 1 + 2 α α ≥ 1 / 2 \alpha\geq1/2 α ≥ 1/2 ϑ ε 1 + 2 α ≤ ϑ ε 2 \vartheta_\varepsilon^{1+2\alpha}\leq\vartheta_\varepsilon^2 ϑ ε 1 + 2 α ≤ ϑ ε 2 ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ]
Proofs of the corollaries
Proof of Corollary 3.1. ∣ I ε − I 0 ∣ = ∣ E [ D ε ] ∣ ≤ E [ ∣ D ε ∣ ] |I^\varepsilon-I^0|=|\mathbb{E}[D^\varepsilon]|\leq\mathbb{E}[|D^\varepsilon|] ∣ I ε − I 0 ∣ = ∣ E [ D ε ] ∣ ≤ E [ ∣ D ε ∣ ] (14) , E [ ∣ D ε ∣ ] ≤ [ f ] α E [ ∣ Z ε ∣ α ] ≤ [ f ] α ( E [ ∣ Z ε ∣ 2 ] ) α / 2 ≤ [ f ] α C 1 α / 2 ε α \mathbb{E}[|D^\varepsilon|]\leq[f]_\alpha\mathbb{E}[|Z^\varepsilon|^\alpha]\leq[f]_\alpha(\mathbb{E}[|Z^\varepsilon|^2])^{\alpha/2}\leq[f]_\alpha C_1^{\alpha/2}\varepsilon^\alpha E [ ∣ D ε ∣ ] ≤ [ f ] α E [ ∣ Z ε ∣ α ] ≤ [ f ] α ( E [ ∣ Z ε ∣ 2 ] ) α /2 ≤ [ f ] α C 1 α /2 ε α (20) , E [ ∣ D ε ∣ ] ≤ TV ( f ) sup u P ( E u ) \mathbb{E}[|D^\varepsilon|]\leq\operatorname{TV}(f)\sup_u\mathbb{P}(E_u) E [ ∣ D ε ∣ ] ≤ TV ( f ) sup u P ( E u ) sup u P ( E u ) ≤ ( ( T + 1 ) C 0 + 2 M K 1 ) ϑ ε \sup_u\mathbb{P}(E_u)\leq((T+1)C_0+2MK_1)\vartheta_\varepsilon sup u P ( E u ) ≤ (( T + 1 ) C 0 + 2 M K 1 ) ϑ ε E [ ∣ D ε ∣ ] \mathbb{E}[|D^\varepsilon|] E [ ∣ D ε ∣ ]
c α , S [ f ] α , S ( E [ ∣ Z ε ∣ 2 α ] ) 1 / 2 P ( ∣ Z ε ∣ > θ ) 1 / 2 ≤ C c α , S [ f ] α , S ϑ ε α ε 2 , c_{\alpha,S}[f]_{\alpha,S}\left(\mathbb{E}[|Z^\varepsilon|^{2\alpha}]\right)^{1/2}\mathbb{P}(|Z^\varepsilon|>\theta)^{1/2}\leq Cc_{\alpha,S}[f]_{\alpha,S}\vartheta_\varepsilon^\alpha\varepsilon^2, c α , S [ f ] α , S ( E [ ∣ Z ε ∣ 2 α ] ) 1/2 P ( ∣ Z ε ∣ > θ ) 1/2 ≤ C c α , S [ f ] α , S ϑ ε α ε 2 , c α , S [ f ] α , S θ α ⋅ 4 ∣ S ∣ M θ , c_{\alpha,S}[f]_{\alpha,S}\theta^\alpha\cdot4|S|M\theta, c α , S [ f ] α , S θ α ⋅ 4∣ S ∣ M θ , and
2 [ f ] α , S θ E [ Υ α − 1 1 { Υ > 2 θ } ] ≤ 2 [ f ] α , S θ ( 1 + 2 ∣ S ∣ M α ) , 2[f]_{\alpha,S}\theta\mathbb{E}\left[\Upsilon^{\alpha-1}1_{\{\Upsilon>2\theta\}}\right]\leq2[f]_{\alpha,S}\theta\left(1+\frac{2|S|M}{\alpha}\right), 2 [ f ] α , S θ E [ Υ α − 1 1 { Υ > 2 θ } ] ≤ 2 [ f ] α , S θ ( 1 + α 2∣ S ∣ M ) , because ∫ 0 1 x α − 1 d x = 1 / α \int_0^1x^{\alpha-1}\,dx=1/\alpha ∫ 0 1 x α − 1 d x = 1/ α ε ≤ ϑ ε ≤ 1 \varepsilon\leq\vartheta_\varepsilon\leq1 ε ≤ ϑ ε ≤ 1 θ = K 4 ϑ ε \theta=K_4\vartheta_\varepsilon θ = K 4 ϑ ε C α , S ′ [ f ] α , S ϑ ε C'_{\alpha,S}[f]_{\alpha,S}\vartheta_\varepsilon C α , S ′ [ f ] α , S ϑ ε
Proof of Corollary 3.2. By Minkowski’s inequality in L 2 L^2 L 2 ∣ σ I ε − σ 0 ∣ = ∣ ∥ F ε − E [ F ε ] ∥ 2 − ∥ F 0 − E [ F 0 ] ∥ 2 ∣ ≤ ∥ D ε − E [ D ε ] ∥ 2 = σ J ε |\sigma_{I^\varepsilon}-\sigma_0|=\left|\|F^\varepsilon-\mathbb{E}[F^\varepsilon]\|_2-\|F^0-\mathbb{E}[F^0]\|_2\right|\leq\|D^\varepsilon-\mathbb{E}[D^\varepsilon]\|_2=\sigma_{J^\varepsilon} ∣ σ I ε − σ 0 ∣ = ∥ F ε − E [ F ε ] ∥ 2 − ∥ F 0 − E [ F 0 ] ∥ 2 ≤ ∥ D ε − E [ D ε ] ∥ 2 = σ J ε R f ( ε ) ≤ σ 0 2 / 4 R_f(\varepsilon)\leq\sigma_0^2/4 R f ( ε ) ≤ σ 0 2 /4 σ J ε ≤ σ 0 / 2 \sigma_{J^\varepsilon}\leq\sigma_0/2 σ J ε ≤ σ 0 /2 σ I ε ≥ σ 0 / 2 \sigma_{I^\varepsilon}\geq\sigma_0/2 σ I ε ≥ σ 0 /2 σ J ε 2 / σ I ε 2 ≤ 4 R f ( ε ) / σ 0 2 \sigma_{J^\varepsilon}^2/\sigma_{I^\varepsilon}^2\leq4R_f(\varepsilon)/\sigma_0^2 σ J ε 2 / σ I ε 2 ≤ 4 R f ( ε ) / σ 0 2 σ K ε 2 / σ I ε 2 = 1 − ( ρ ε ) 2 \sigma_{K^\varepsilon}^2/\sigma_{I^\varepsilon}^2=1-(\rho^\varepsilon)^2 σ K ε 2 / σ I ε 2 = 1 − ( ρ ε ) 2 (13) , the inequality σ K ε ≤ σ J ε \sigma_{K^\varepsilon}\leq\sigma_{J^\varepsilon} σ K ε ≤ σ J ε (12) , and (13) also gives σ 0 2 ( 1 − ε ) 2 = σ J ε 2 − σ K ε 2 ≤ σ J ε 2 \sigma_0^2(1-\varepsilon)^2=\sigma_{J^\varepsilon}^2-\sigma_{K^\varepsilon}^2\leq\sigma_{J^\varepsilon}^2 σ 0 2 ( 1 − ε ) 2 = σ J ε 2 − σ K ε 2 ≤ σ J ε 2
Numerical experiments
Test problems and predicted exponents
To test the exponents of § 3 , measure the variance reduction, and check that time discretization does not affect these conclusions, we take σ = 2 \sigma=\sqrt{2} σ = 2 v 0 = 0.5 v_0=0.5 v 0 = 0.5 T = 1 T=1 T = 1 v f = 2 v_f=2 v f = 2 Δ = 1 \Delta=1 Δ = 1 γ = 1 \gamma=1 γ = 1 ℓ = 0.5 \ell=0.5 ℓ = 0.5 d ′ = 1 d'=1 d ′ = 1 A = K = r = 1 A=K=r=1 A = K = r = 1 s = σ = 2 s=\sigma=\sqrt{2} s = σ = 2 [12 ] reduce to the identity. The observables are f i n d f_{\mathrm{ind}} f ind f α f_\alpha f α α ∈ { 1 / 4 , 1 / 3 , 1 / 2 , 3 / 4 , 1 } \alpha\in\{1/4,1/3,1/2,3/4,1\} α ∈ { 1/4 , 1/3 , 1/2 , 3/4 , 1 } f s m f_{\mathrm{sm}} f sm Figure 1 (b), f α f_\alpha f α α ∈ { 0.1 , 0.2 , 0.3 , 0.4 , 0.6 , 0.7 , 0.8 , 0.9 } \alpha\in\{0.1,0.2,0.3,0.4,0.6,0.7,0.8,0.9\} α ∈ { 0.1 , 0.2 , 0.3 , 0.4 , 0.6 , 0.7 , 0.8 , 0.9 }
Figure 1. (a) Local slopes r i r_i r i D 2 D_2 D 2 f 1 f_1 f 1 ε i \varepsilon_i ε i D 2 D_2 D 2 ε ≤ 0.05 \varepsilon\le0.05 ε ≤ 0.05 α \alpha α f i n d f_{\mathrm{ind}} f ind α = 0 \alpha= 0 α = 0 f s m f_{\mathrm{sm}} f sm min { 2 , 1 + 2 α } \min\{2,1+2\alpha\} min { 2 , 1 + 2 α } max { 2 α , 2 / 3 } \max\{2\alpha,2/3\} max { 2 α , 2/3 } (14) .
By Theorem 3.1, the exponent of D 2 ( ε ) ≜ E [ ( D ε ) 2 ] D_2(\varepsilon)\triangleq\mathbb{E}[(D^\varepsilon)^2] D 2 ( ε ) ≜ E [( D ε ) 2 ] 1 1 1 f i n d f_{\mathrm{ind}} f ind min { 2 , 1 + 2 α } \min\{2,1+2\alpha\} min { 2 , 1 + 2 α } f α f_\alpha f α 2 2 2 f 1 f_1 f 1 f s m f_{\mathrm{sm}} f sm f i n d f_{\mathrm{ind}} f ind f α f_\alpha f α α < 1 \alpha<1 α < 1 max { 2 α , 2 / 3 } \max\{2\alpha,2/3\} max { 2 α , 2/3 } (14) and Avikainen’s inequality [44 ] , so that the experiment separates the two. They are exponents of upper bounds, but we expect them to be attained, because ∣ Z ε ( T ) ∣ |Z^\varepsilon(T)| ∣ Z ε ( T ) ∣ ε \varepsilon ε p T ( ± v f ) > 0 p_T(\pm v_f)>0 p T ( ± v f ) > 0 (14) gives the exponent 2 2 2 E [ ∣ Z ε ( T ) ∣ 2 ] \mathbb{E}[|Z^\varepsilon(T)|^2] E [ ∣ Z ε ( T ) ∣ 2 ] Proposition B.2 (iii) the exponent 2 2 2 E [ sup t ≤ T ∣ Z ε ( t ) ∣ 2 ] \mathbb{E}[\sup_{t\leq T}|Z^\varepsilon(t)|^2] E [ sup t ≤ T ∣ Z ε ( t ) ∣ 2 ]
Coupled time discretization
Let δ t = T / n \delta t=T/n δ t = T / n ζ 0 , ζ 1 , … \zeta_0,\zeta_1,\ldots ζ 0 , ζ 1 , … N ( 0 , I d ′ ) \mathcal{N}(0,I_{d'}) N ( 0 , I d ′ ) η 0 ∼ N ( 0 , C ) \eta_0\sim\mathcal{N}(0,C) η 0 ∼ N ( 0 , C )
η m + 1 = η m − δ t ε 2 A η m + δ t ε K ζ m , δ N m ε ≜ σ δ t ε r ⋅ η m , δ N m 0 ≜ σ δ t c T K ζ m . \eta_{m+1}=\eta_m-\frac{\delta t}{\varepsilon^2}A\eta_m+\frac{\sqrt{\delta t}}{\varepsilon}K\zeta_m,\qquad\delta N_m^\varepsilon\triangleq\frac{\sigma\delta t}{\varepsilon}r\cdot\eta_m,\qquad\delta N_m^0\triangleq\sigma\sqrt{\delta t}\,c^\mathsf{T}K\zeta_m. η m + 1 = η m − ε 2 δ t A η m + ε δ t K ζ m , δ N m ε ≜ ε σ δ t r ⋅ η m , δ N m 0 ≜ σ δ t c T K ζ m . Multiplying the first relation by ε σ r T A − 1 \varepsilon\sigma r^\mathsf{T}A^{-1} ε σ r T A − 1 δ N m ε = δ N m 0 − ε σ c T ( η m + 1 − η m ) \delta N_m^\varepsilon=\delta N_m^0-\varepsilon\sigma c^\mathsf{T}(\eta_{m+1}-\eta_m) δ N m ε = δ N m 0 − ε σ c T ( η m + 1 − η m ) (5) ,
h m ≜ ∑ j < m ( δ N j ε − δ N j 0 ) = − σ ε c T ( η m − η 0 ) , m ≥ 0. (22) h_m\triangleq\sum_{j<m}(\delta N_j^\varepsilon-\delta N_j^0)=-\sigma\varepsilon c^\mathsf{T}(\eta_m-\eta_0),\qquad m\geq0. \tag*{(22)} h m ≜ j < m ∑ ( δ N j ε − δ N j 0 ) = − σ ε c T ( η m − η 0 ) , m ≥ 0. (22) The velocities are advanced by the backward Euler (proximal) step for U U U
V m + 1 = prox δ t U ( V m + δ N m ) = soft Δ δ t ( V m + ℓ δ t + δ N m ) 1 + γ δ t , soft τ ( x ) ≜ sgn ( x ) ( ∣ x ∣ − τ ) + . (23) V_{m+1}=\operatorname{prox}_{\delta t U}(V_m+\delta N_m)=\frac{\operatorname{soft}_{\Delta\delta t}(V_m+\ell\delta t+\delta N_m)}{1+\gamma\delta t},\qquad\operatorname{soft}_{\tau}(x)\triangleq\operatorname{sgn}(x)(|x|-\tau)_+. \tag*{(23)} V m + 1 = prox δ t U ( V m + δ N m ) = 1 + γ δ t soft Δ δ t ( V m + ℓ δ t + δ N m ) , soft τ ( x ) ≜ sgn ( x ) ( ∣ x ∣ − τ ) + . (23) with δ N m = δ N m ε \delta N_m=\delta N_m^\varepsilon δ N m = δ N m ε V ε V^\varepsilon V ε δ N m = δ N m 0 \delta N_m=\delta N_m^0 δ N m = δ N m 0 V 0 V^0 V 0 V 0 = v 0 V_0=v_0 V 0 = v 0 V m + 1 = V m + δ N m + δ t ( b ˉ ( V m + 1 ) − k m + 1 ) V_{m+1}=V_m+\delta N_m+\delta t(\bar b(V_{m+1})-k_{m+1}) V m + 1 = V m + δ N m + δ t ( b ˉ ( V m + 1 ) − k m + 1 ) k m + 1 ∈ ∂ φ ( V m + 1 ) k_{m+1}\in\partial\varphi(V_{m+1}) k m + 1 ∈ ∂ φ ( V m + 1 ) V 0 V^0 V 0 U U U [47 ] ; for V ε V^\varepsilon V ε [50 ] . We take n = ⌈ T / min { 10 − 3 , ε 2 / 50 } ⌉ n=\lceil T/\min\{10^{-3},\varepsilon^2/50\}\rceil n = ⌈ T / min { 1 0 − 3 , ε 2 /50 }⌉ n δ t = T n\delta t=T n δ t = T δ t A / ε 2 ≤ 1 / 50 \delta t A/\varepsilon^2\leq1/50 δ t A / ε 2 ≤ 1/50 ε \varepsilon ε δ t \delta t δ t Proposition 5.1 also applies are detailed in Appendix § 10 .
Proposition 5.1 (Discrete comparison principle). Let ( V m ε ) (V_m^\varepsilon) ( V m ε ) ( V m 0 ) (V_m^0) ( V m 0 ) (23) from the same ( η 0 , ζ 0 , ζ 1 , … ) (\eta_0,\zeta_0,\zeta_1,\ldots) ( η 0 , ζ 0 , ζ 1 , … ) n ≥ 0 n\geq0 n ≥ 0
max m ≤ n ∣ V m ε − V m 0 ∣ ≤ 2 max m ≤ n ∣ h m ∣ ≤ 4 k 0 ε max m ≤ n ∣ η m ∣ . (24) \max_{m\leq n}|V_m^\varepsilon-V_m^0|\leq2\max_{m\leq n}|h_m|\leq4k_0\varepsilon\max_{m\leq n}|\eta_m|. \tag*{(24)} m ≤ n max ∣ V m ε − V m 0 ∣ ≤ 2 m ≤ n max ∣ h m ∣ ≤ 4 k 0 ε m ≤ n max ∣ η m ∣. (24) Proof. Let N m ≜ ∑ j < m δ N j N_m\triangleq\sum_{j<m}\delta N_j N m ≜ ∑ j < m δ N j Y m ≜ V m − N m Y_m\triangleq V_m-N_m Y m ≜ V m − N m Λ m ≜ Y m ε − Y m 0 \Lambda_m\triangleq Y_m^\varepsilon-Y_m^0 Λ m ≜ Y m ε − Y m 0 Λ 0 = 0 \Lambda_0=0 Λ 0 = 0 V m ε − V m 0 = Λ m + h m V_m^\varepsilon-V_m^0=\Lambda_m+h_m V m ε − V m 0 = Λ m + h m H m ≜ max j ≤ m ∣ h j ∣ H_m\triangleq\max_{j\leq m}|h_j| H m ≜ max j ≤ m ∣ h j ∣ ∣ Λ m ∣ ≤ H m |\Lambda_m|\leq H_m ∣ Λ m ∣ ≤ H m (24) ; the second follows from (22) . Here Y m + 1 = Y m − δ t g m + 1 Y_{m+1}=Y_m-\delta t\,g_{m+1} Y m + 1 = Y m − δ t g m + 1 g m + 1 ≔ k m + 1 − b ˉ ( V m + 1 ) ∈ ∂ U ( V m + 1 ) g_{m+1}\coloneqq k_{m+1}-\bar b(V_{m+1})\in\partial U(V_{m+1}) g m + 1 : = k m + 1 − b ˉ ( V m + 1 ) ∈ ∂ U ( V m + 1 ) ∣ Λ m ∣ ≤ H m |\Lambda_m|\le H_m ∣ Λ m ∣ ≤ H m Λ m + 1 > H m + 1 \Lambda_{m+1}>H_{m+1} Λ m + 1 > H m + 1 V m + 1 ε − V m + 1 o = Λ m + 1 + h m + 1 > 0 V_{m+1}^{\varepsilon}-V_{m+1}^{\mathrm{o}}=\Lambda_{m+1}+h_{m+1}>0 V m + 1 ε − V m + 1 o = Λ m + 1 + h m + 1 > 0 g m + 1 ε ≥ g m + 1 o g_{m+1}^{\varepsilon}\ge g_{m+1}^{\mathrm{o}} g m + 1 ε ≥ g m + 1 o ∂ U \partial U ∂ U Λ m + 1 = Λ m − δ t ( g m + 1 ε − g m + 1 o ) ≤ Λ m ≤ H m ≤ H m + 1 \Lambda_{m+1}=\Lambda_m-\delta t(g_{m+1}^{\varepsilon}-g_{m+1}^{\mathrm{o}})\le\Lambda_m\le H_m\le H_{m+1} Λ m + 1 = Λ m − δ t ( g m + 1 ε − g m + 1 o ) ≤ Λ m ≤ H m ≤ H m + 1 Λ m + 1 ≥ − H m + 1 \Lambda_{m+1}\ge-H_{m+1} Λ m + 1 ≥ − H m + 1
Thus the coupling error of the simulated pair is bounded uniformly in δ t \delta t δ t δ t 1 / 4 \delta t^{1/4} δ t 1/4 V o V^{\mathrm{o}} V o [47 ] ) enters only the bias of the estimators, which the refinement test below monitors. The first inequality in (24) is checked on every simulated path.
Reference values and statistical methodology
Reference values. The value I 0 = ∫ R f p T I^{0}=\int_{\mathbb{R}}f\,p_T I 0 = ∫ R f p T [12 ] , truncated at 40 modes, in regimes (II)–(III). This density is cross-validated against the numerical inversion of the Laplace transform of [13 ] , against a finite-volume solution of the Fokker–Planck equation, and by simulation (Appendix § 10 and Table 3 ): the spectral and Laplace representations agree to 10 − 15 10^{-15} 1 0 − 15 2 ⋅ 10 − 7 2\cdot10^{-7} 2 ⋅ 1 0 − 7 I 0 I^{0} I 0 J ^ N ε \widehat J_N^\varepsilon J N ε K ^ N ε \widehat K_N^\varepsilon K N ε σ J ε N − 1 / 2 \sigma_{J^\varepsilon}N^{-1/2} σ J ε N − 1/2
Table 3. Reference values I 0 = E [ f ( V 0 ( T ) ) ] I^{0} = \mathbb{E}[f(V^{0}(T))] I 0 = E [ f ( V 0 ( T ))] p T p_T p T I 0 I^{0} I 0 p T p_T p T [12 (2.10)] on a grid of [ − 6 , 6 ] [-6,6] [ − 6 , 6 ] [13 ] and the finite-volume solution at the five test points; the largest discrepancy between I 0 I^{0} I 0 sup p T \sup p_T sup p T
Measured quantities. (1) E [ ∣ Z ε ( T ) ∣ 2 ] \mathbb{E}[|Z^\varepsilon(T)|^2] E [ ∣ Z ε ( T ) ∣ 2 ] E [ max m ≤ n ∣ Z m ε ∣ 2 ] \mathbb{E}[\max_{m\le n}|Z_m^\varepsilon|^2] E [ max m ≤ n ∣ Z m ε ∣ 2 ] E [ sup t ≤ T ∣ Z ε ( t ) ∣ 2 ] \mathbb{E}[\sup_{t\le T}|Z^\varepsilon(t)|^2] E [ sup t ≤ T ∣ Z ε ( t ) ∣ 2 ] (14) and Proposition B.2 . (2) D 2 ( ε ) D_2(\varepsilon) D 2 ( ε ) r i r_i r i log D 2 \log D_2 log D 2 log ε \log\varepsilon log ε § 10 ); the smooth anchor calibrates the finite-grid deficit shared by all observables. (3) The variance ratios σ J ε 2 / σ ε 2 \sigma_{J^\varepsilon}^2/\sigma_\varepsilon^2 σ J ε 2 / σ ε 2 σ K ε 2 / σ ε 2 = 1 − ( ρ ε ) 2 \sigma_{K^\varepsilon}^2/\sigma_\varepsilon^2=1-(\rho^\varepsilon)^2 σ K ε 2 / σ ε 2 = 1 − ( ρ ε ) 2 λ ^ N ε \widehat\lambda_N^\varepsilon λ N ε E [ D ε ] \mathbb{E}[D^\varepsilon] E [ D ε ] I ε − I 0 I^\varepsilon-I^0 I ε − I 0 max m ∣ Z m ε ∣ / ( 2 max m ∣ h m ∣ ) \max_m|Z_m^\varepsilon|/(2\max_m|h_m|) max m ∣ Z m ε ∣/ ( 2 max m ∣ h m ∣ ) Proposition 5.1 , and the standardized difference between the sample mean of f ( V n 0 ) f(V_n^0) f ( V n 0 ) I 0 I^0 I 0 V 0 V^0 V 0
Grid and sampling. The rate study uses thirteen values of ε \varepsilon ε ε ∈ { 0.6 , 0.8 , 1 } \varepsilon\in\{0.6,0.8,1\} ε ∈ { 0.6 , 0.8 , 1 } ε \varepsilon ε N = 2.4 ⋅ 10 5 N=2.4\cdot10^5 N = 2.4 ⋅ 1 0 5 ε ≥ 0.05 \varepsilon\ge0.05 ε ≥ 0.05 N = 4.8 ⋅ 10 5 N=4.8\cdot10^5 N = 4.8 ⋅ 1 0 5 B = 12 \mathcal{B}=12 B = 12 D 2 D_2 D 2 (11) are computed from all N N N λ ^ N ε \widehat\lambda_N^\varepsilon λ N ε log D 2 \log D_2 log D 2 log ε \log\varepsilon log ε ε ≤ 0.05 \varepsilon\le0.05 ε ≤ 0.05 ε ≤ 0.2 \varepsilon\le0.2 ε ≤ 0.2 ∣ E [ D ε ] ∣ |\mathbb{E}[D^\varepsilon]| ∣ E [ D ε ] ∣ § 10 ). The refinement test repeats the cells ε ∈ { 0.2 , 0.1 , 0.05 } \varepsilon\in\{0.2,0.1,0.05\} ε ∈ { 0.2 , 0.1 , 0.05 } n = ⌈ T / min { 10 − 3 , ε 2 / 400 } ⌉ n=\lceil T/\min\{10^{-3},\varepsilon^2/400\}\rceil n = ⌈ T / min { 1 0 − 3 , ε 2 /400 }⌉ N = 2.4 ⋅ 10 5 N=2.4\cdot10^5 N = 2.4 ⋅ 1 0 5 4.8 ⋅ 10 11 4.8\cdot10^{11} 4.8 ⋅ 1 0 11
Results
Coupling errors. The fitted exponents of E [ ∣ Z ε ( T ) ∣ 2 ] \mathbb{E}[|Z^\varepsilon(T)|^2] E [ ∣ Z ε ( T ) ∣ 2 ] Table 1 ), in agreement with the exponent 2 of (14) . Those of E [ max m ∣ Z m ε ∣ 2 ] \mathbb{E}[\max_m|Z_m^\varepsilon|^2] E [ max m ∣ Z m ε ∣ 2 ] Proposition B.2 (iii): the maximum of the OU process over the T / ε 2 T/\varepsilon^2 T / ε 2 ( log ( 1 / ε ) ) 1 / 2 (\log(1/\varepsilon))^{1/2} ( log ( 1/ ε ) ) 1/2 [51 ] , and the local exponent 2 − 1 / ( a + log ( 1 / ε ) ) 2-1/(a+\log(1/\varepsilon)) 2 − 1/ ( a + log ( 1/ ε )) ε 2 ( a + log ( 1 / ε ) ) \varepsilon^2(a+\log(1/\varepsilon)) ε 2 ( a + log ( 1/ ε )) a ≥ 0 a\ge0 a ≥ 0 [ 0.02 , 0.05 ] [0.02,0.05] [ 0.02 , 0.05 ]
Table 1. Fitted exponents of D 2 ( ε ) \mathcal{D}_2(\varepsilon) D 2 ( ε ) ε ≤ 0.05 \varepsilon\le0.05 ε ≤ 0.05 ∣ E [ D ε ] ∣ |\mathbb{E}[D^\varepsilon]| ∣ E [ D ε ] ∣ ε ≤ 0.2 \varepsilon\le0.2 ε ≤ 0.2 max { 2 α , 2 / 3 } \max\{2\alpha, 2/3\} max { 2 α , 2/3 } [27 Lemma 6.1] the exponent 2 for f s m f_{\mathrm{sm}} f sm
Exponents of D 2 D_2 D 2 For f i n d f_{\mathrm{ind}} f ind f 1 f_1 f 1 f s m f_{\mathrm{sm}} f sm Table 1 ); in particular, the exponent for the crossing indicator is 1.08(4), 0.83(10), and 1.02(10) in regimes (I)–(III). For f α f_\alpha f α 1 / 4 ≤ α ≤ 3 / 4 1/4 \le\alpha\le3/4 1/4 ≤ α ≤ 3/4 min { 2 , 1 + 2 α } \min\{2,1+2\alpha\} min { 2 , 1 + 2 α } α = 1 / 2 \alpha= 1/2 α = 1/2 Figure 1 (b)). Such deficits are expected on a finite window. Heuristically, the first correction to the leading term of D 2 ( ε ) D_2(\varepsilon) D 2 ( ε ) ε ∣ 1 − 2 α ∣ \varepsilon^{\lvert1-2\alpha\rvert} ε ∣ 1 − 2 α ∣ α \alpha α 1 / 2 1/2 1/2 α = 1 / 2 \alpha= 1/2 α = 1/2 Z ε ( T ) Z^\varepsilon(T) Z ε ( T ) V 0 ( T ) V^0(T) V 0 ( T ) § 10 ), gives on the same window exponents below min { 2 , 1 + 2 α } \min\{2,1+2\alpha\} min { 2 , 1 + 2 α } α \alpha α f i n d f_{\mathrm{ind}} f ind f 1 f_1 f 1 f s m f_{\mathrm{sm}} f sm max { 2 α , 2 / 3 } \max\{2\alpha,2/3\} max { 2 α , 2/3 } f i n d f_{\mathrm{ind}} f ind D 2 D_2 D 2 (14) alone can guarantee, as Theorem 3.1 asserts.
Bias. The coupled bias is resolved for all 21 pairs of observable and regime, with fitted exponents between 1.74 and 2.08. These exceed the exponent 1 of Corollary 3.1 and are close to the exponent 2 known for smooth observables [27 Lemma 6.1] , which suggests that the bias bounds of Corollary 3.1(b)–(c) are not sharp for these examples. The white-noise approximation is nevertheless inaccurate at moderate ε \varepsilon ε f i n d f_{\mathrm{ind}} f ind ε = 0.2 \varepsilon= 0.2 ε = 0.2 ∣ E [ D ε ] ∣ \lvert\mathbb{E}[D^\varepsilon]\rvert ∣ E [ D ε ]∣ I ε I^\varepsilon I ε Table 2 ), a bias that the control variate estimators remove.
Table 2. Crossing indicator f i n d f_{\mathrm{ind}} f ind K ^ N ε \widehat{K}^{\varepsilon}_{N} K N ε I ε I^\varepsilon I ε E [ D ε ] = I ε − I 0 \mathbb{E}[D^\varepsilon] = I^\varepsilon-I^0 E [ D ε ] = I ε − I 0 σ I ε \sigma_{I^\varepsilon} σ I ε λ ^ N ε \widehat{\lambda}^{\varepsilon}_{N} λ N ε K ^ N ε \widehat{K}^{\varepsilon}_{N} K N ε 2 σ K ε 2 / σ I ε 2 2\sigma^2_{K^\varepsilon}/\sigma^2_{I^\varepsilon} 2 σ K ε 2 / σ I ε 2 ε = 1 \varepsilon= 1 ε = 1 V ε V^\varepsilon V ε
Variance reduction. For the crossing indicator (Table 2 ), σ K ε 2 / σ I ε 2 \sigma_{K^\varepsilon}^2/\sigma_{I^\varepsilon}^2 σ K ε 2 / σ I ε 2 ε = 1 \varepsilon= 1 ε = 1 ε = 0.02 \varepsilon= 0.02 ε = 0.02 ε \varepsilon ε ε ≤ 0.1 \varepsilon\le0.1 ε ≤ 0.1 λ ^ N ε \widehat{\lambda}^\varepsilon_N λ N ε ε = 0.02 \varepsilon= 0.02 ε = 0.02 K ^ N ε \widehat{K}^\varepsilon_N K N ε ε = 1 \varepsilon= 1 ε = 1 F ε F^\varepsilon F ε F 0 F^0 F 0 V ε V^\varepsilon V ε J ^ N ε \widehat{J}^\varepsilon_N J N ε σ J ε 2 / σ I ε 2 \sigma_{J^\varepsilon}^2/\sigma_{I^\varepsilon}^2 σ J ε 2 / σ I ε 2 λ ^ N ε ≈ 0 \widehat{\lambda}^\varepsilon_N \approx0 λ N ε ≈ 0 K ^ N ε \widehat{K}^\varepsilon_N K N ε σ K ε 2 / σ I ε 2 \sigma_{K^\varepsilon}^2/\sigma_{I^\varepsilon}^2 σ K ε 2 / σ I ε 2 ε = 0.02 \varepsilon= 0.02 ε = 0.02 9.4 ⋅ 10 − 4 9.4 \cdot10^{-4} 9.4 ⋅ 1 0 − 4 1.4 ⋅ 10 − 3 1.4 \cdot10^{-3} 1.4 ⋅ 1 0 − 3
Conclusion
For the inertial dry-friction inclusion driven by colored noise, rate estimates are derived for the control variate estimators of [27 ] with non-smooth observables. The analysis rests on two structural properties of the scalar velocity inclusion. The first is a pathwise comparison principle, which yields 2-Lipschitz continuity of the solution map for the uniform norm and survives time discretization. The second is an explicit density bound for the white-noise terminal velocity, obtained by a Girsanov comparison with pure dry friction. These properties give Gaussian-tail coupling errors and yield O ( ε 2 α ) O(\varepsilon^{2\alpha}) O ( ε 2 α ) α \alpha α O ( ε ) O(\varepsilon) O ( ε ) O ( ε 1 + α ) O(\varepsilon^{1+\alpha}) O ( ε 1 + α ) α \alpha α O ( ε min { 2 , 1 + 2 α } ) O(\varepsilon^{\min\{2,1+2\alpha\}}) O ( ε m i n { 2 , 1 + 2 α } ) α \alpha α
Numerical experiments in the three regimes support these rates: the fitted exponents of the second moment agree with the theorem for the crossing indicator and the Lipschitz observables, agree with a model computation of the pre-asymptotic corrections for the severity family, and exceed those that the mean-square coupling bound alone would give. The experiments also indicate that the bias of the white-noise approximation is of order ε 2 \varepsilon^{2} ε 2
Code and data availability
No experimental data were used in this study. The Python code that produces the numerical results, tables and figures of § 5 and Appendix § 10 , together with the generated data files, is available at https://github.com/Lihong-Guo/Dry-friction.git for research purposes.
Use of generative AI
The author used DeepSeek and Claude (Anthropic) for language editing, mathematical assistance, and numerical scripting. The author reviewed all outputs and take full responsibility for the final content.
Appendices
Auxiliary lemmas
No sticking under white noise
The following lemma is a standard consequence of the occupation-times formula [52 Chapter VI, Corollary 1.6] . The absence of sticking under white noise is noted in [12 ] .
Lemma A.1 (No sticking under white noise). Assume that Assumptions A1, A2, and A3 hold. Then V 0 V^0 V 0 ⟨ V 0 ⟩ t = ς 2 t \langle V^0\rangle_t = \varsigma^2t ⟨ V 0 ⟩ t = ς 2 t Leb { t ∈ [ 0 , T ] : V 0 ( t ) = 0 } = 0 \operatorname{Leb}\{t \in[0,T] : V^0(t) = 0\} = 0 Leb { t ∈ [ 0 , T ] : V 0 ( t ) = 0 } = 0 k 0 ( t ) = Δ sgn ( V 0 ( t ) ) k^0(t) = \Delta\operatorname{sgn}(V^0(t)) k 0 ( t ) = Δ sgn ( V 0 ( t )) t ∈ [ 0 , T ] t \in[0,T] t ∈ [ 0 , T ] b ( v ) ≜ b ˉ ( v ) − Δ sgn ( v ) b(v) \triangleq\bar{b}(v) - \Delta\operatorname{sgn}(v) b ( v ) ≜ b ˉ ( v ) − Δ sgn ( v )
V 0 ( t ) = v 0 + ∫ 0 t b ( V 0 ( u ) ) d u + ς B ( t ) , t ∈ [ 0 , T ] . (25) V^0(t) = v_0 + \int_0^t b(V^0(u))\,\mathrm{d}u + \varsigma B(t), \qquad t \in[0,T]. \tag*{(25)} V 0 ( t ) = v 0 + ∫ 0 t b ( V 0 ( u )) d u + ς B ( t ) , t ∈ [ 0 , T ] . (25) Proof. By (8) , V 0 ( t ) = v 0 + ∫ 0 t ( b ˉ ( V 0 ) − k 0 ) d u + ς B ( t ) V^0(t) = v_0 + \int_0^t(\bar{b}(V^0)-k^0)\,\mathrm{d}u+\varsigma B(t) V 0 ( t ) = v 0 + ∫ 0 t ( b ˉ ( V 0 ) − k 0 ) d u + ς B ( t ) V 0 ( t ) − v 0 − ς B ( t ) V^0(t)-v_0-\varsigma B(t) V 0 ( t ) − v 0 − ς B ( t ) V 0 V^0 V 0 ⟨ V 0 ⟩ t = ς 2 t \langle V^0\rangle_t = \varsigma^2t ⟨ V 0 ⟩ t = ς 2 t ς 2 ∫ 0 T 1 { V 0 ( t ) = 0 } d t = ∫ R 1 { 0 } ( a ) L T a ( V 0 ) d a = 0 \varsigma^2\int_0^T \mathbf{1}_{\{V^0(t)=0\}}\,\mathrm{d}t = \int_{\mathbb{R}}\mathbf{1}_{\{0\}}(a)L^a_T(V^0)\,\mathrm{d}a = 0 ς 2 ∫ 0 T 1 { V 0 ( t ) = 0 } d t = ∫ R 1 { 0 } ( a ) L T a ( V 0 ) d a = 0 L a ( V 0 ) L^a(V^0) L a ( V 0 ) V 0 V^0 V 0 a a a ς > 0 \varsigma> 0 ς > 0 Leb { t : V 0 ( t ) = 0 } = 0 \operatorname{Leb}\{t : V^0(t) = 0\} = 0 Leb { t : V 0 ( t ) = 0 } = 0 t t t V 0 ( t ) ≠ 0 V^0(t) \ne0 V 0 ( t ) = 0 ∂ φ ( V 0 ( t ) ) \partial\varphi(V^0(t)) ∂ φ ( V 0 ( t )) { Δ sgn ( V 0 ( t ) ) } \{\Delta\operatorname{sgn}(V^0(t))\} { Δ sgn ( V 0 ( t ))} k 0 ( t ) = Δ sgn ( V 0 ( t ) ) k^0(t) = \Delta\operatorname{sgn}(V^0(t)) k 0 ( t ) = Δ sgn ( V 0 ( t )) t t t
Maximal inequality for the fast OU process
The following two lemmas are standard: the expected supremum of an OU process over a time interval of length t t t ( log t ) 1 / 2 (\log t)^{1/2} ( log t ) 1/2 [51 ] . We include their short proofs because the constants C 0 C_0 C 0 c 0 c_0 c 0 Proposition B.2 .
Lemma A.2. Let η ~ \widetilde{\eta} η d η ~ = − A η ~ d u + K d W ^ \mathrm{d}\widetilde{\eta}=-A\widetilde{\eta}\,\mathrm{d}u+K\,\mathrm{d}\widehat{W} d η = − A η d u + K d W W ^ \widehat{W} W d ′ d' d ′ η ~ ( 0 ) ∼ N ( 0 , C ) \widetilde{\eta}(0)\sim\mathcal{N}(0,C) η ( 0 ) ∼ N ( 0 , C ) W ^ \widehat{W} W a A ≔ max u ∈ [ 0 , 1 ] ∥ e − A u ∥ a_A\coloneqq\max_{u\in[0,1]}\lVert e^{-Au}\rVert a A : = max u ∈ [ 0 , 1 ] ∥ e − A u ∥ b A , K ≔ ∥ K ∥ + max u ∈ [ 0 , 1 ] ∥ A e − A u K ∥ b_{A,K}\coloneqq\lVert K\rVert+\max_{u\in[0,1]}\lVert Ae^{-Au}K\rVert b A , K : = ∥ K ∥ + max u ∈ [ 0 , 1 ] ∥ A e − A u K ∥ c C ≔ max i C i i c_C\coloneqq\max_i C_{ii} c C : = max i C ii Y 1 ≔ sup u ∈ [ 0 , 1 ] ∣ η ~ ( u ) ∣ Y_1\coloneqq\sup_{u\in[0,1]}|\widetilde{\eta}(u)| Y 1 : = sup u ∈ [ 0 , 1 ] ∣ η ( u ) ∣ y ≥ 0 y\geq0 y ≥ 0
P ( Y 1 > y ) ≤ d e − y 2 / ( 8 a A 2 d c C ) + 2 d ′ e − y 2 / ( 8 b A , K 2 d ′ ) ≤ C 0 e − c 0 y 2 , \mathbb{P}(Y_1>y)\leq d e^{-y^2/(8a_A^2dc_C)}+2d'e^{-y^2/(8b_{A,K}^2d')}\leq C_0e^{-c_0y^2}, P ( Y 1 > y ) ≤ d e − y 2 / ( 8 a A 2 d c C ) + 2 d ′ e − y 2 / ( 8 b A , K 2 d ′ ) ≤ C 0 e − c 0 y 2 , with C 0 ≔ d + 2 d ′ C_0\coloneqq d+2d' C 0 : = d + 2 d ′ c 0 ≔ min { ( 8 a A 2 d c C ) − 1 , ( 8 b A , K 2 d ′ ) − 1 } c_0\coloneqq\min\{(8a_A^2dc_C)^{-1},(8b_{A,K}^2d')^{-1}\} c 0 : = min {( 8 a A 2 d c C ) − 1 , ( 8 b A , K 2 d ′ ) − 1 } K = 0 K=0 K = 0 η ~ ≡ 0 \widetilde{\eta}\equiv0 η ≡ 0
Proof. By the variation-of-constants formula, η ~ ( u ) = e − A u η ~ ( 0 ) + ∫ 0 u e − A ( u − r ) K d W ^ r \widetilde{\eta}(u)=e^{-Au}\widetilde{\eta}(0)+\int_0^u e^{-A(u-r)}K\,\mathrm{d}\widehat{W}_r η ( u ) = e − A u η ( 0 ) + ∫ 0 u e − A ( u − r ) K d W r r ↦ e − A ( u − r ) K r\mapsto e^{-A(u-r)}K r ↦ e − A ( u − r ) K C 1 C^1 C 1 ∫ 0 u e − A ( u − r ) K d W ^ r = K W ^ u − ∫ 0 u A e − A ( u − r ) K W ^ r d r \int_0^u e^{-A(u-r)}K\,\mathrm{d}\widehat{W}_r=K\widehat{W}_u-\int_0^u Ae^{-A(u-r)}K\widehat{W}_r\,\mathrm{d}r ∫ 0 u e − A ( u − r ) K d W r = K W u − ∫ 0 u A e − A ( u − r ) K W r d r u u u u ∈ [ 0 , 1 ] u\in[0,1] u ∈ [ 0 , 1 ] Y 1 ≤ a A ∣ η ~ ( 0 ) ∣ + b A , K sup r ≤ 1 ∣ W ^ r ∣ Y_1\leq a_A|\widetilde{\eta}(0)|+b_{A,K}\sup_{r\leq1}|\widehat{W}_r| Y 1 ≤ a A ∣ η ( 0 ) ∣ + b A , K sup r ≤ 1 ∣ W r ∣ X ∈ R d X\in\mathbb{R}^d X ∈ R d C C C P ( ∣ X ∣ > z ) ≤ ∑ i P ( ∣ X i ∣ > z / d ) ≤ d e − z 2 / ( 2 d c C ) \mathbb{P}(|X|>z)\leq\sum_i\mathbb{P}(|X_i|>z/\sqrt{d})\leq d e^{-z^2/(2dc_C)} P ( ∣ X ∣ > z ) ≤ ∑ i P ( ∣ X i ∣ > z / d ) ≤ d e − z 2 / ( 2 d c C ) P ( ∣ N ( 0 , ς 2 ) ∣ > x ) ≤ e − x 2 / ( 2 ς 2 ) \mathbb{P}(|\mathcal{N}(0,\varsigma^2)|>x)\leq e^{-x^2/(2\varsigma^2)} P ( ∣ N ( 0 , ς 2 ) ∣ > x ) ≤ e − x 2 / ( 2 ς 2 ) P ( sup r ≤ 1 ∣ W ^ r j ∣ > x ) ≤ 2 P ( ∣ W ^ 1 j ∣ > x ) ≤ 2 e − x 2 / 2 \mathbb{P}(\sup_{r\leq1}|\widehat{W}^j_r|>x)\leq2\mathbb{P}(|\widehat{W}^j_1|>x)\leq2e^{-x^2/2} P ( sup r ≤ 1 ∣ W r j ∣ > x ) ≤ 2 P ( ∣ W 1 j ∣ > x ) ≤ 2 e − x 2 /2 P ( sup r ≤ 1 ∣ W ^ r ∣ > z ) ≤ 2 d ′ e − z 2 / ( 2 d ′ ) \mathbb{P}(\sup_{r\leq1}|\widehat{W}_r|>z)\leq2d'e^{-z^2/(2d')} P ( sup r ≤ 1 ∣ W r ∣ > z ) ≤ 2 d ′ e − z 2 / ( 2 d ′ ) P ( Y 1 > y ) ≤ P ( a A ∣ η ~ ( 0 ) ∣ > y / 2 ) + P ( b A , K sup r ≤ 1 ∣ W ^ r ∣ > y / 2 ) \mathbb{P}(Y_1>y)\leq\mathbb{P}(a_A|\widetilde{\eta}(0)|>y/2)+\mathbb{P}(b_{A,K}\sup_{r\leq1}|\widehat{W}_r|>y/2) P ( Y 1 > y ) ≤ P ( a A ∣ η ( 0 ) ∣ > y /2 ) + P ( b A , K sup r ≤ 1 ∣ W r ∣ > y /2 )
Lemma A.3 (Maximal inequality for the fast OU process). Assume that Assumption A2 holds, and let C 0 , c 0 C_0,c_0 C 0 , c 0 ε > 0 \varepsilon>0 ε > 0 t > 0 t>0 t > 0 y ≥ 0 y\geq0 y ≥ 0
P ( sup u ≤ t ∣ η ε ( u ) ∣ > y ) ≤ C 0 ⌈ t ε 2 ⌉ e − c 0 y 2 . (26) \mathbb{P}\left(\sup_{u\leq t}|\eta^\varepsilon(u)|>y\right)\leq C_0\left\lceil\frac{t}{\varepsilon^2}\right\rceil e^{-c_0y^2}. \tag*{(26)} P ( u ≤ t sup ∣ η ε ( u ) ∣ > y ) ≤ C 0 ⌈ ε 2 t ⌉ e − c 0 y 2 . (26) Consequently, for every p > 0 p>0 p > 0 L p L_p L p T T T E [ sup u ≤ T ∣ η ε ( u ) ∣ p ] ≤ L p ( 1 + log ( 1 / ε ) ) p / 2 \mathbb{E}[\sup_{u\leq T}|\eta^\varepsilon(u)|^p]\leq L_p(1+\log(1/\varepsilon))^{p/2} E [ sup u ≤ T ∣ η ε ( u ) ∣ p ] ≤ L p ( 1 + log ( 1/ ε ) ) p /2 ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ]
Proof. With η ^ ( u ) = η ε ( ε 2 u ) \widehat{\eta}(u)=\eta^\varepsilon(\varepsilon^2u) η ( u ) = η ε ( ε 2 u ) § 2.2 ), sup u ≤ t ∣ η ε ( u ) ∣ = sup u ≤ t / ε 2 ∣ η ^ ( u ) ∣ \sup_{u\leq t}|\eta^\varepsilon(u)|=\sup_{u\leq t/\varepsilon^2}|\widehat{\eta}(u)| sup u ≤ t ∣ η ε ( u ) ∣ = sup u ≤ t / ε 2 ∣ η ( u ) ∣ [ 0 , t / ε 2 ] [0,t/\varepsilon^2] [ 0 , t / ε 2 ] m ≔ ⌈ t / ε 2 ⌉ m\coloneqq\lceil t/\varepsilon^2\rceil m : = ⌈ t / ε 2 ⌉ [ j , j + 1 ] [j,j+1] [ j , j + 1 ] 0 ≤ j ≤ m − 1 0\leq j\leq m-1 0 ≤ j ≤ m − 1 η ^ \widehat{\eta} η sup u ∈ [ j , j + 1 ] ∣ η ^ ( u ) ∣ \sup_{u\in[j,j+1]}|\widehat{\eta}(u)| sup u ∈ [ j , j + 1 ] ∣ η ( u ) ∣ Y 1 Y_1 Y 1 j j j
For the moments, let Y ≔ sup u ≤ T ∣ η ε ( u ) ∣ Y\coloneqq\sup_{u\leq T}|\eta^\varepsilon(u)| Y : = sup u ≤ T ∣ η ε ( u ) ∣ m ≔ ⌈ T / ε 2 ⌉ m\coloneqq\lceil T/\varepsilon^2\rceil m : = ⌈ T / ε 2 ⌉ y 0 ≔ ( log m / c 0 ) 1 / 2 y_0\coloneqq(\log m/c_0)^{1/2} y 0 : = ( log m / c 0 ) 1/2 y ≥ y 0 y\geq y_0 y ≥ y 0 m C 0 e − c 0 y 2 ≤ C 0 e − c 0 ( y 2 − y 0 2 ) ≤ C 0 e − c 0 ( y − y 0 ) 2 mC_0e^{-c_0y^2}\leq C_0e^{-c_0(y^2-y_0^2)}\leq C_0e^{-c_0(y-y_0)^2} m C 0 e − c 0 y 2 ≤ C 0 e − c 0 ( y 2 − y 0 2 ) ≤ C 0 e − c 0 ( y − y 0 ) 2 p ≥ 1 p\geq1 p ≥ 1
E [ Y p ] = ∫ 0 ∞ p y p − 1 P ( Y > y ) d y ≤ y 0 p + C 0 p ∫ 0 ∞ ( z + y 0 ) p − 1 e − c 0 z 2 d z ≤ L p ( 1 + y 0 ) p , \mathbb{E}[Y^p]=\int_0^\infty py^{p-1}\mathbb{P}(Y>y)\,\mathrm{d}y\leq y_0^p+C_0p\int_0^\infty(z+y_0)^{p-1}e^{-c_0z^2}\,\mathrm{d}z\leq L_p(1+y_0)^p, E [ Y p ] = ∫ 0 ∞ p y p − 1 P ( Y > y ) d y ≤ y 0 p + C 0 p ∫ 0 ∞ ( z + y 0 ) p − 1 e − c 0 z 2 d z ≤ L p ( 1 + y 0 ) p , using ( z + y 0 ) p − 1 ≤ 2 ( p − 2 ) + ( z p − 1 + y 0 p − 1 ) (z+y_0)^{p-1}\leq2^{(p-2)_+}(z^{p-1}+y_0^{p-1}) ( z + y 0 ) p − 1 ≤ 2 ( p − 2 ) + ( z p − 1 + y 0 p − 1 ) ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ] m ≤ ( T + 1 ) ε − 2 m\leq(T+1)\varepsilon^{-2} m ≤ ( T + 1 ) ε − 2 y 0 2 ≤ c 0 − 1 ( log ( T + 1 ) + 2 log ( 1 / ε ) ) ≤ C ( 1 + log ( 1 / ε ) ) y_0^2\leq c_0^{-1}(\log(T+1)+2\log(1/\varepsilon))\leq C(1+\log(1/\varepsilon)) y 0 2 ≤ c 0 − 1 ( log ( T + 1 ) + 2 log ( 1/ ε )) ≤ C ( 1 + log ( 1/ ε )) p ∈ ( 0 , 1 ) p\in(0,1) p ∈ ( 0 , 1 )
Two lemmas on observables
Both lemmas are elementary. Lemma A.4 is the pathwise inequality underlying Avikainen-type estimates [44 ] , stated for the class B V J \mathrm{BV}_J BV J
Lemma A.4 (Crossing inequality). Let f ∈ B V J f \in\mathrm{BV}_{J} f ∈ BV J μ f \mu_f μ f (10) . Then, for all x , y ∈ R x,y \in\mathbb{R} x , y ∈ R
∣ f ( x ) − f ( y ) ∣ ≤ ∣ μ f ∣ ( [ [ x , y ] ] ) , ( f ( x ) − f ( y ) ) 2 ≤ T V ( f ) ∣ μ f ∣ ( [ [ x , y ] ] ) . |f(x)-f(y)| \leq|\mu_f|([[x,y]]), \qquad(f(x)-f(y))^2 \leq\mathrm{TV}(f)|\mu_f|([[x,y]]). ∣ f ( x ) − f ( y ) ∣ ≤ ∣ μ f ∣ ([[ x , y ]]) , ( f ( x ) − f ( y ) ) 2 ≤ TV ( f ) ∣ μ f ∣ ([[ x , y ]]) . Proof. Let J − ≜ { x ∈ R : f ( x ) ≠ f + ( x ) } J^- \triangleq\{x \in\mathbb{R}: f(x) \ne f_+(x)\} J − ≜ { x ∈ R : f ( x ) = f + ( x )} f ( x ) = f − ( x ) f(x)=f_-(x) f ( x ) = f − ( x ) x ∈ J − x \in J^- x ∈ J − f ( x ) = f + ( x ) f(x)=f_+(x) f ( x ) = f + ( x ) x ∉ J − x \notin J^- x ∈ / J − χ ( u , y ) ≜ 1 { u < y } \chi(u,y) \triangleq\mathbf{1}_{\{u<y\}} χ ( u , y ) ≜ 1 { u < y } u ∈ J − u \in J^- u ∈ J − χ ( u , y ) ≜ 1 { u ≤ y } \chi(u,y) \triangleq\mathbf{1}_{\{u\leq y\}} χ ( u , y ) ≜ 1 { u ≤ y } χ ( u , y ) = 1 { u ≤ y } − 1 J − ( y ) 1 { u = y } \chi(u,y)=\mathbf{1}_{\{u\leq y\}}-\mathbf{1}_{J^-}(y)\mathbf{1}_{\{u=y\}} χ ( u , y ) = 1 { u ≤ y } − 1 J − ( y ) 1 { u = y } (10) ,
f ( − ∞ ) + ∫ R χ ( u , y ) μ f ( d u ) = f + ( y ) − 1 J − ( y ) ( f + ( y ) − f − ( y ) ) = f ( y ) ( y ∈ R ) . f(-\infty)+\int_{\mathbb{R}}\chi(u,y)\,\mu_f(\mathrm{d}u)=f_+(y)-\mathbf{1}_{J^-}(y)(f_+(y)-f_-(y))=f(y)\qquad(y\in\mathbb{R}). f ( − ∞ ) + ∫ R χ ( u , y ) μ f ( d u ) = f + ( y ) − 1 J − ( y ) ( f + ( y ) − f − ( y )) = f ( y ) ( y ∈ R ) . Hence f ( x ) − f ( y ) = ∫ R ( χ ( u , x ) − χ ( u , y ) ) μ f ( d u ) f(x)-f(y)=\int_{\mathbb{R}}(\chi(u,x)-\chi(u,y))\,\mu_f(\mathrm{d}u) f ( x ) − f ( y ) = ∫ R ( χ ( u , x ) − χ ( u , y )) μ f ( d u ) u u u y ↦ χ ( u , y ) y\mapsto\chi(u,y) y ↦ χ ( u , y ) ( u , ∞ ) (u,\infty) ( u , ∞ ) [ u , ∞ ) [u,\infty) [ u , ∞ ) χ ( u , x ) ≠ χ ( u , y ) \chi(u,x)\ne\chi(u,y) χ ( u , x ) = χ ( u , y ) u ∈ [ [ x , y ] ] u\in[[x,y]] u ∈ [[ x , y ]]
∣ f ( x ) − f ( y ) ∣ ≤ ∫ R 1 [ [ x , y ] ] ( u ) ∣ μ f ∣ ( d u ) = ∣ μ f ∣ ( [ [ x , y ] ] ) , |f(x)-f(y)|\leq\int_{\mathbb{R}}\mathbf{1}_{[[x,y]]}(u)|\mu_f|(\mathrm{d}u)=|\mu_f|([[x,y]]), ∣ f ( x ) − f ( y ) ∣ ≤ ∫ R 1 [[ x , y ]] ( u ) ∣ μ f ∣ ( d u ) = ∣ μ f ∣ ([[ x , y ]]) , and the second inequality follows from ∣ μ f ∣ ( [ [ x , y ] ] ) ≤ ∣ μ f ∣ ( R ) ≤ T V ( f ) |\mu_f|([[x,y]])\leq|\mu_f|(\mathbb{R})\leq\mathrm{TV}(f) ∣ μ f ∣ ([[ x , y ]]) ≤ ∣ μ f ∣ ( R ) ≤ TV ( f )
Lemma A.5 (Cusp class). Let f ∈ K S α f\in\mathcal{K}^{\alpha}_{S} f ∈ K S α
(i) if [ [ x , y ] ] ∩ S = ∅ [[x,y]]\cap S=\varnothing [[ x , y ]] ∩ S = ∅ ∣ f ( x ) − f ( y ) ∣ ≤ [ f ] α , S dist ( [ [ x , y ] ] , S ) α − 1 ∣ x − y ∣ |f(x)-f(y)|\leq[f]_{\alpha,S}\operatorname{dist}([[x,y]],S)^{\alpha-1}|x-y| ∣ f ( x ) − f ( y ) ∣ ≤ [ f ] α , S dist ([[ x , y ]] , S ) α − 1 ∣ x − y ∣
(ii) f ∈ C 0 , α f\in C^{0,\alpha} f ∈ C 0 , α [ f ] α ≤ c α , S [ f ] α , S [f]_{\alpha}\leq c_{\alpha,S}[f]_{\alpha,S} [ f ] α ≤ c α , S [ f ] α , S c α , S ≜ 2 2 − α ∣ S ∣ / α c_{\alpha,S}\triangleq2^{2-\alpha}|S|/\alpha c α , S ≜ 2 2 − α ∣ S ∣/ α
Proof. Since f f f ∣ f ( x ) − f ( y ) ∣ ≤ ∫ [ x , y ] ∣ f ′ ( u ) ∣ d u ≤ [ f ] α , S ∫ [ x , y ] dist ( u , S ) α − 1 d u |f(x)-f(y)|\leq\int_{[x,y]}|f'(u)|\mathrm{d}u\leq[f]_{\alpha,S}\int_{[x,y]}\operatorname{dist}(u,S)^{\alpha-1}\mathrm{d}u ∣ f ( x ) − f ( y ) ∣ ≤ ∫ [ x , y ] ∣ f ′ ( u ) ∣ d u ≤ [ f ] α , S ∫ [ x , y ] dist ( u , S ) α − 1 d u dist ( u , S ) α − 1 ≤ dist ( [ [ x , y ] ] , S ) α − 1 \operatorname{dist}(u,S)^{\alpha-1}\leq\operatorname{dist}([[x,y]],S)^{\alpha-1} dist ( u , S ) α − 1 ≤ dist ([[ x , y ]] , S ) α − 1 u ∈ [ x , y ] u\in[x,y] u ∈ [ x , y ] α − 1 ≤ 0 \alpha-1\leq0 α − 1 ≤ 0 dist ( u , S ) α − 1 = max ς ∈ S ∣ u − ς ∣ α − 1 ≤ ∑ ς ∈ S ∣ u − ς ∣ α − 1 \operatorname{dist}(u,S)^{\alpha-1}=\max_{\varsigma\in S}|u-\varsigma|^{\alpha-1}\leq\sum_{\varsigma\in S}|u-\varsigma|^{\alpha-1} dist ( u , S ) α − 1 = max ς ∈ S ∣ u − ς ∣ α − 1 ≤ ∑ ς ∈ S ∣ u − ς ∣ α − 1 J J J h h h ς ∈ R \varsigma\in\mathbb{R} ς ∈ R J J J ∣ u − ς ∣ ≤ h / 2 |u-\varsigma|\leq h/2 ∣ u − ς ∣ ≤ h /2
∫ J ∣ u − ς ∣ α − 1 d u ≤ 2 ∫ 0 h / 2 z α − 1 d z + ( h / 2 ) α − 1 h = 2 1 − α α h α + 2 1 − α h α ≤ 2 2 − α α h α . \int_J|u-\varsigma|^{\alpha-1}\mathrm{d}u\leq2\int_0^{h/2}z^{\alpha-1}\mathrm{d}z+(h/2)^{\alpha-1}h=\frac{2^{1-\alpha}}{\alpha}h^\alpha+2^{1-\alpha}h^\alpha\leq\frac{2^{2-\alpha}}{\alpha}h^\alpha. ∫ J ∣ u − ς ∣ α − 1 d u ≤ 2 ∫ 0 h /2 z α − 1 d z + ( h /2 ) α − 1 h = α 2 1 − α h α + 2 1 − α h α ≤ α 2 2 − α h α . Applying this with J = [ [ x , y ] ] J=[[x,y]] J = [[ x , y ]] h = ∣ x − y ∣ h=|x-y| h = ∣ x − y ∣ ς ∈ S \varsigma\in S ς ∈ S
The law of the white-noise velocity
To the best of our knowledge, Propositions B.1 and B.2 are new. Proposition B.3 gives an explicit bound on the density of V 0 ( T ) V^0(T) V 0 ( T )
Proposition B.1 (Pathwise comparison principle). Let γ ≥ 0 \gamma\geq0 γ ≥ 0 ℓ ∈ R \ell\in\mathbb{R} ℓ ∈ R w , w ′ ∈ C 0 0 ( [ 0 , T ] ) w,w'\in C^0_0([0,T]) w , w ′ ∈ C 0 0 ([ 0 , T ]) t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ]
sup u ≤ t ∣ S ( w ) ( u ) − S ( w ′ ) ( u ) ∣ ≤ 2 sup u ≤ t ∣ w ( u ) − w ′ ( u ) ∣ . (27) \sup_{u\leq t}|\mathcal{S}(w)(u)-\mathcal{S}(w')(u)|\leq2\sup_{u\leq t}|w(u)-w'(u)|. \tag*{(27)} u ≤ t sup ∣ S ( w ) ( u ) − S ( w ′ ) ( u ) ∣ ≤ 2 u ≤ t sup ∣ w ( u ) − w ′ ( u ) ∣. (27) In particular, S \mathcal{S} S 2 2 2 ( C 0 0 ( [ 0 , T ] ) , ∥ ⋅ ∥ ∞ ) (C^0_0([0,T]),\|\cdot\|_\infty) ( C 0 0 ([ 0 , T ]) , ∥ ⋅ ∥ ∞ )
Proof. Let ( v , k ) (v,k) ( v , k ) ( v ′ , k ′ ) (v',k') ( v ′ , k ′ ) (7) for w w w w ′ w' w ′ g ≜ k − b ˉ ( v ) g\triangleq k-\bar b(v) g ≜ k − b ˉ ( v ) g ′ ≜ k ′ − b ˉ ( v ′ ) g'\triangleq k'-\bar b(v') g ′ ≜ k ′ − b ˉ ( v ′ ) [ 0 , T ] [0,T] [ 0 , T ] h ≜ w − w ′ h\triangleq w-w' h ≜ w − w ′ (8) , the function Λ ≜ ( v − w ) − ( v ′ − w ′ ) = − ∫ 0 ⋅ ( g − g ′ ) d u \Lambda\triangleq(v-w)-(v'-w')=-\int_0^\cdot(g-g')\mathrm{d}u Λ ≜ ( v − w ) − ( v ′ − w ′ ) = − ∫ 0 ⋅ ( g − g ′ ) d u Λ ( 0 ) = 0 \Lambda(0)=0 Λ ( 0 ) = 0 Λ ′ = − ( g − g ′ ) \Lambda'=-(g-g') Λ ′ = − ( g − g ′ ) v − v ′ = Λ + h v-v'=\Lambda+h v − v ′ = Λ + h u u u
( g ( u ) − g ′ ( u ) ) ( v ( u ) − v ′ ( u ) ) = ( k ( u ) − k ′ ( u ) ) ( v ( u ) − v ′ ( u ) ) + γ ( v ( u ) − v ′ ( u ) ) 2 ≥ 0 , (g(u)-g'(u))(v(u)-v'(u))=(k(u)-k'(u))(v(u)-v'(u))+\gamma(v(u)-v'(u))^2\geq0, ( g ( u ) − g ′ ( u )) ( v ( u ) − v ′ ( u )) = ( k ( u ) − k ′ ( u )) ( v ( u ) − v ′ ( u )) + γ ( v ( u ) − v ′ ( u ) ) 2 ≥ 0 , by the monotonicity of ∂ φ \partial\varphi ∂ φ g ( u ) ≥ g ′ ( u ) g(u)\geq g'(u) g ( u ) ≥ g ′ ( u ) v ( u ) > v ′ ( u ) v(u)>v'(u) v ( u ) > v ′ ( u ) t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ] H ≜ sup u ≤ t ∣ h ( u ) ∣ H\triangleq\sup_{u\leq t}|h(u)| H ≜ sup u ≤ t ∣ h ( u ) ∣ Λ ( t 1 ) > H \Lambda(t_1)>H Λ ( t 1 ) > H t 1 ≤ t t_1\leq t t 1 ≤ t τ ≜ sup { u ∈ [ 0 , t 1 ] : Λ ( u ) ≤ H } \tau\triangleq\sup\{u\in[0,t_1]:\Lambda(u)\leq H\} τ ≜ sup { u ∈ [ 0 , t 1 ] : Λ ( u ) ≤ H } 0 0 0 Λ ( τ ) ≤ H \Lambda(\tau)\leq H Λ ( τ ) ≤ H τ < t 1 \tau<t_1 τ < t 1 Λ > H \Lambda>H Λ > H ( τ , t 1 ] (\tau,t_1] ( τ , t 1 ] ( τ , t 1 ] (\tau,t_1] ( τ , t 1 ] v − v ′ = Λ + h > H − ∣ h ∣ ≥ 0 v-v'=\Lambda+h>H-|h|\geq0 v − v ′ = Λ + h > H − ∣ h ∣ ≥ 0 g ≥ g ′ g\geq g' g ≥ g ′ Λ ′ ≤ 0 \Lambda'\leq0 Λ ′ ≤ 0 Λ ( t 1 ) ≤ Λ ( τ ) ≤ H \Lambda(t_1)\leq\Lambda(\tau)\leq H Λ ( t 1 ) ≤ Λ ( τ ) ≤ H Λ ≤ H \Lambda\leq H Λ ≤ H [ 0 , t ] [0,t] [ 0 , t ] Λ ≥ − H \Lambda\geq-H Λ ≥ − H ∣ v ( u ) − v ′ ( u ) ∣ ≤ ∣ Λ ( u ) ∣ + ∣ h ( u ) ∣ ≤ 2 H |v(u)-v'(u)|\leq|\Lambda(u)|+|h(u)|\leq2H ∣ v ( u ) − v ′ ( u ) ∣ ≤ ∣Λ ( u ) ∣ + ∣ h ( u ) ∣ ≤ 2 H u ≤ t u\leq t u ≤ t
Proposition B.2 (Coupling error). Assume that Assumptions A1 and A2 (but not necessarily Assumption A3) hold, and let C 0 , c 0 > 0 C_0,c_0>0 C 0 , c 0 > 0 A , K , d , d ′ A,K,d,d' A , K , d , d ′
(i) almost surely, sup t ≤ T ∣ Z ε ( t ) ∣ ≤ 2 sup t ≤ T ∣ h ε ( t ) ∣ ≤ 4 κ 0 ε sup t ≤ T ∣ η ε ( t ) ∣ \sup_{t\leq T}|Z^\varepsilon(t)|\leq2\sup_{t\leq T}|h^\varepsilon(t)|\leq4\kappa_0\varepsilon\sup_{t\leq T}|\eta^\varepsilon(t)| sup t ≤ T ∣ Z ε ( t ) ∣ ≤ 2 sup t ≤ T ∣ h ε ( t ) ∣ ≤ 4 κ 0 ε sup t ≤ T ∣ η ε ( t ) ∣
(ii) for all x ≥ 0 x\geq0 x ≥ 0 P ( sup t ≤ T ∣ Z ε ( t ) ∣ > x ) ≤ ⌈ T / ε 2 ⌉ C 0 exp ( − c 0 x 2 / ( 16 κ 0 2 ε 2 ) ) \mathbb{P}\left(\sup_{t\leq T}|Z^\varepsilon(t)|>x\right)\leq\lceil T/\varepsilon^2\rceil C_0\exp\left(-c_0x^2/(16\kappa_0^2\varepsilon^2)\right) P ( sup t ≤ T ∣ Z ε ( t ) ∣ > x ) ≤ ⌈ T / ε 2 ⌉ C 0 exp ( − c 0 x 2 / ( 16 κ 0 2 ε 2 ) )
(iii) for every p > 0 p>0 p > 0 C p C_p C p E [ sup t ≤ T ∣ Z ε ( t ) ∣ p ] ≤ C p ϑ ε p \mathbb{E}\left[\sup_{t\leq T}|Z^\varepsilon(t)|^p\right]\leq C_p\vartheta_\varepsilon^p E [ sup t ≤ T ∣ Z ε ( t ) ∣ p ] ≤ C p ϑ ε p ε ∈ ( 0 , 1 ] \varepsilon\in(0,1] ε ∈ ( 0 , 1 ]
(iv) for every q > 0 q>0 q > 0 K q ≜ 4 κ 0 ( ( q + 2 ) / c 0 ) 1 / 2 K_q\triangleq4\kappa_0((q+2)/c_0)^{1/2} K q ≜ 4 κ 0 (( q + 2 ) / c 0 ) 1/2
P ( sup t ≤ T ∣ Z ε ( t ) ∣ > K q ϑ ε ) ≤ ( T + 1 ) C 0 ε q , ε ∈ ( 0 , 1 ] . \mathbb{P}\left(\sup_{t\leq T}|Z^\varepsilon(t)|>K_q\vartheta_\varepsilon\right)\leq(T+1)C_0\varepsilon^q,\qquad\varepsilon\in(0,1]. P ( t ≤ T sup ∣ Z ε ( t ) ∣ > K q ϑ ε ) ≤ ( T + 1 ) C 0 ε q , ε ∈ ( 0 , 1 ] . Proof. (i) By (9) , V ε = S ( N ε ) V^\varepsilon=S(N^\varepsilon) V ε = S ( N ε ) V 0 = S ( N 0 ) V^0=S(N^0) V 0 = S ( N 0 ) N ε , N 0 ∈ C 0 ( [ 0 , T ] ) N^\varepsilon,N^0\in C^0([0,T]) N ε , N 0 ∈ C 0 ([ 0 , T ]) (6) the second. (ii) follows from (i) and (A.2), and (iii) from (i) and the moment bound of Lemma A.3. (iv) For x = K q ϑ ε x=K_q\vartheta_\varepsilon x = K q ϑ ε c 0 x 2 / ( 16 κ 0 2 ε 2 ) = ( q + 2 ) ( 1 + log ( 1 / ε ) ) c_0x^2/(16\kappa_0^2\varepsilon^2)=(q+2)(1+\log(1/\varepsilon)) c 0 x 2 / ( 16 κ 0 2 ε 2 ) = ( q + 2 ) ( 1 + log ( 1/ ε )) ⌈ T / ε 2 ⌉ ≤ ( T + 1 ) ε − 2 \lceil T/\varepsilon^2\rceil\leq(T+1)\varepsilon^{-2} ⌈ T / ε 2 ⌉ ≤ ( T + 1 ) ε − 2 ε ≤ 1 \varepsilon\leq1 ε ≤ 1 P ( ⋅ ) ≤ ( T + 1 ) ε − 2 C 0 e − ( q + 2 ) ε q + 2 ≤ ( T + 1 ) C 0 ε q \mathbb{P}(\cdot)\leq(T+1)\varepsilon^{-2}C_0e^{-(q+2)}\varepsilon^{q+2}\leq(T+1)C_0\varepsilon^q P ( ⋅ ) ≤ ( T + 1 ) ε − 2 C 0 e − ( q + 2 ) ε q + 2 ≤ ( T + 1 ) C 0 ε q
Proposition B.2 strengthens, with a rate, the convergence in probability of [27 Proposition 5.1(2)] ; the factor ( 1 + log ( 1 / ε ) ) 1 / 2 (1+\log(1/\varepsilon))^{1/2} ( 1 + log ( 1/ ε ) ) 1/2 T / ε 2 T/\varepsilon^2 T / ε 2 [51 ] .
Proposition B.3 (Density of the white-noise velocity). Assume that Assumptions A1, A2, and A3 hold, and let U , U ∗ U,U_* U , U ∗ (3) . Then the law of V 0 ( T ) V^0(T) V 0 ( T ) p T p_T p T p T ≤ p ˉ T p_T\leq\bar p_T p T ≤ p ˉ T p ˉ T \bar p_T p ˉ T
p ˉ T ( v ) ≜ e γ T 2 + U ( v 0 ) − U ( v ) s 2 [ e − ( v − v 0 ) 2 2 s 2 T 1 2 π s 2 T + Δ s 2 e Δ 2 T 2 s 2 − Δ ( ∣ v ∣ + ∣ v 0 ∣ ) s 2 Φ ( ∣ v ∣ + ∣ v 0 ∣ − Δ T s T ) ] , v ∈ R . (28) \bar p_T(v)\triangleq e^{\frac{\gamma T}{2}+\frac{U(v_0)-U(v)}{s^2}}\left[e^{-\frac{(v-v_0)^2}{2s^2T}}\frac{1}{\sqrt{2\pi s^2T}}+\frac{\Delta}{s^2}e^{\frac{\Delta^2T}{2s^2}-\frac{\Delta(|v|+|v_0|)}{s^2}}\Phi\left(\frac{|v|+|v_0|-\Delta T}{s\sqrt{T}}\right)\right],\qquad v\in\mathbb{R}. \tag*{(28)} p ˉ T ( v ) ≜ e 2 γ T + s 2 U ( v 0 ) − U ( v ) [ e − 2 s 2 T ( v − v 0 ) 2 2 π s 2 T 1 + s 2 Δ e 2 s 2 Δ 2 T − s 2 Δ ( ∣ v ∣ + ∣ v 0 ∣ ) Φ ( s T ∣ v ∣ + ∣ v 0 ∣ − Δ T ) ] , v ∈ R . (28) In particular,
M ≜ ess sup v ∈ R p T ( v ) ≤ e γ T 2 + U ( v 0 ) − U ∗ s 2 [ 1 2 π s 2 T + Δ s 2 e Δ 2 T 2 s 2 ] < ∞ . (29) M\triangleq\operatorname*{ess\,sup}_{v\in\mathbb{R}}p_T(v)\leq e^{\frac{\gamma T}{2}+\frac{U(v_0)-U_*}{s^2}}\left[\frac{1}{\sqrt{2\pi s^2T}}+\frac{\Delta}{s^2}e^{\frac{\Delta^2T}{2s^2}}\right]<\infty. \tag*{(29)} M ≜ v ∈ R ess sup p T ( v ) ≤ e 2 γ T + s 2 U ( v 0 ) − U ∗ [ 2 π s 2 T 1 + s 2 Δ e 2 s 2 Δ 2 T ] < ∞. (29) Proof. Let b b b U 0 ( v ) ≜ γ 2 v 2 − ℓ v U_0(v)\triangleq\frac{\gamma}{2}v^2-\ell v U 0 ( v ) ≜ 2 γ v 2 − ℓ v U = U 0 + Δ ∣ ⋅ ∣ U=U_0+\Delta|\cdot| U = U 0 + Δ∣ ⋅ ∣ b ˉ = − U 0 ′ \bar b=-U_0' b ˉ = − U 0 ′ E ⊂ R E\subset\mathbb{R} E ⊂ R
P ( V 0 ( T ) ∈ E ) ≤ ∫ E p ˉ T ( v ) d v . (30) \mathbb{P}(V^0(T)\in E)\leq\int_E\bar p_T(v)\,\mathrm{d}v. \tag*{(30)} P ( V 0 ( T ) ∈ E ) ≤ ∫ E p ˉ T ( v ) d v . (30) and deduce the proposition from (B.4) in Step 4.
Step 1 (Girsanov transformation to pure dry friction). Let Θ u ≜ b ˉ ( V 0 ( u ) ) / s \Theta_u\triangleq\bar b(V^0(u))/s Θ u ≜ b ˉ ( V 0 ( u )) / s ∣ b ˉ ( v ) ∣ ≤ γ ∣ v ∣ + ∣ ℓ ∣ |\bar b(v)|\leq\gamma|v|+|\ell| ∣ b ˉ ( v ) ∣ ≤ γ ∣ v ∣ + ∣ ℓ ∣ [ 0 , u ] [0,u] [ 0 , u ] ∣ V 0 ( u ) ∣ ≤ ∥ S ( 0 ) ∥ ∞ + 2 s sup r ≤ u ∣ B ( r ) ∣ |V^0(u)|\leq\|S(0)\|_\infty+2s\sup_{r\leq u}|B(r)| ∣ V 0 ( u ) ∣ ≤ ∥ S ( 0 ) ∥ ∞ + 2 s sup r ≤ u ∣ B ( r ) ∣ C Θ C_\Theta C Θ ∣ Θ u ∣ ≤ C Θ ( 1 + sup r ≤ u ∣ B ( r ) ∣ ) |\Theta_u|\leq C_\Theta(1+\sup_{r\leq u}|B(r)|) ∣ Θ u ∣ ≤ C Θ ( 1 + sup r ≤ u ∣ B ( r ) ∣ ) V 0 V^0 V 0 B B B [53 Chapter 3, Corollary 5.16] shows that
Z t ≜ exp ( − ∫ 0 t Θ u d B ( u ) − 1 2 ∫ 0 t Θ u 2 d u ) , t ∈ [ 0 , T ] , Z_t\triangleq\exp\left(-\int_0^t\Theta_u\,\mathrm{d}B(u)-\frac{1}{2}\int_0^t\Theta_u^2\,\mathrm{d}u\right),\qquad t\in[0,T], Z t ≜ exp ( − ∫ 0 t Θ u d B ( u ) − 2 1 ∫ 0 t Θ u 2 d u ) , t ∈ [ 0 , T ] , is a martingale. Define Q ≜ Z T ⋅ P \mathbb{Q}\triangleq Z_T\cdot\mathbb{P} Q ≜ Z T ⋅ P F T \mathcal{F}_T F T [53 Chapter 3, Theorem 5.1] , B ~ ( t ) ≜ B ( t ) + ∫ 0 t Θ u d u \widetilde B(t)\triangleq B(t)+\int_0^t\Theta_u\,\mathrm{d}u B ( t ) ≜ B ( t ) + ∫ 0 t Θ u d u Q \mathbb{Q} Q [ 0 , T ] [0,T] [ 0 , T ]
V 0 ( t ) = v 0 − Δ ∫ 0 t sgn ( V 0 ( u ) ) d u + s B ~ ( t ) , t ∈ [ 0 , T ] . V^0(t)=v_0-\Delta\int_0^t\operatorname{sgn}(V^0(u))\,\mathrm{d}u+s\widetilde B(t),\qquad t\in[0,T]. V 0 ( t ) = v 0 − Δ ∫ 0 t sgn ( V 0 ( u )) d u + s B ( t ) , t ∈ [ 0 , T ] . Thus, under Q \mathbb{Q} Q V 0 V^0 V 0 s s s [53 Chapter 5, Proposition 3.10] ; the value of sgn ( 0 ) \operatorname{sgn}(0) sgn ( 0 ) Lemma A.1 . Hence, under Q \mathbb{Q} Q V 0 ( T ) V^0(T) V 0 ( T ) [12 (2.10)] ,
p T ( I ) ( v ) ≔ e − Δ 2 T 2 s 2 − Δ ( ∣ v ∣ − ∣ v 0 ∣ ) s 2 e − ( v − v 0 ) 2 2 s 2 T 2 π s 2 T + Δ s 2 e − 2 Δ ∣ v ∣ s 2 Φ ( ∣ v ∣ + ∣ v 0 ∣ − Δ T s T ) , v ∈ R . (31) p_T^{(I)}(v) \coloneqq e^{-\frac{\Delta^2T}{2s^2}-\frac{\Delta(|v|-|v_0|)}{s^2}}\frac{e^{-\frac{(v-v_0)^2}{2s^2T}}}{\sqrt{2\pi s^2T}}+\frac{\Delta}{s^2}e^{-\frac{2\Delta|v|}{s^2}}\Phi\left(\frac{|v|+|v_0|-\Delta T}{s\sqrt{T}}\right),\qquad v\in\mathbb{R}. \tag*{(31)} p T ( I ) ( v ) : = e − 2 s 2 Δ 2 T − s 2 Δ ( ∣ v ∣ − ∣ v 0 ∣ ) 2 π s 2 T e − 2 s 2 T ( v − v 0 ) 2 + s 2 Δ e − s 2 2Δ∣ v ∣ Φ ( s T ∣ v ∣ + ∣ v 0 ∣ − Δ T ) , v ∈ R . (31) Since Z T > 0 Z_T>0 Z T > 0 P ≪ Q \mathbb{P}\ll\mathbb{Q} P ≪ Q s d B ~ ( u ) = d V 0 ( u ) + Δ sgn ( V 0 ( u ) ) d u s\,\mathrm{d}\widetilde{B}(u)=\mathrm{d}V^0(u)+\Delta\operatorname{sgn}(V^0(u))\,\mathrm{d}u s d B ( u ) = d V 0 ( u ) + Δ sgn ( V 0 ( u )) d u
d P d Q = Z T − 1 = exp ( ∫ 0 T Θ u d B ~ ( u ) − 1 2 ∫ 0 T Θ u 2 d u ) = exp ( 1 s 2 ∫ 0 T b ‾ ( V 0 ) d V 0 + 1 s 2 ∫ 0 T ( Δ b ‾ ( V 0 ) sgn ( V 0 ) − 1 2 b ‾ [ V 0 ] 2 ) d u ) . \begin{aligned}
\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}=Z_T^{-1}=\exp\left(\int_0^T\Theta_u\,\mathrm{d}\widetilde{B}(u)-\frac{1}{2}\int_0^T\Theta_u^2\,\mathrm{d}u\right) \\
&=\exp\left(\frac{1}{s^2}\int_0^T\overline{b}(V^0)\,\mathrm{d}V^0+\frac{1}{s^2}\int_0^T\left(\Delta\overline{b}(V^0)\operatorname{sgn}(V^0)-\frac{1}{2}\overline{b}[V^0]^2\right)\mathrm{d}u\right).
\end{aligned} d Q d P = Z T − 1 = exp ( ∫ 0 T Θ u d B ( u ) − 2 1 ∫ 0 T Θ u 2 d u ) = exp ( s 2 1 ∫ 0 T b ( V 0 ) d V 0 + s 2 1 ∫ 0 T ( Δ b ( V 0 ) sgn ( V 0 ) − 2 1 b [ V 0 ] 2 ) d u ) . Step 2 (Itô’s formula). Since U 0 ′ ′ = γ U_0^{\prime\prime}=\gamma U 0 ′′ = γ ⟨ V 0 ⟩ t = s 2 t \langle V^0\rangle_t=s^2t ⟨ V 0 ⟩ t = s 2 t ∫ 0 T b ‾ ( V 0 ) d V 0 = U 0 ( v 0 ) − U 0 ( V 0 ( T ) ) + γ s 2 T / 2 \int_0^T\overline{b}(V^0)\,\mathrm{d}V^0=U_0(v_0)-U_0(V^0(T))+\gamma s^2T/2 ∫ 0 T b ( V 0 ) d V 0 = U 0 ( v 0 ) − U 0 ( V 0 ( T )) + γ s 2 T /2 Δ b ‾ sgn − b ‾ 2 / 2 ≤ Δ ∣ b ‾ ∣ − b ‾ 2 / 2 ≤ Δ 2 / 2 \Delta\overline{b}\operatorname{sgn}-\overline{b}^2/2\leq\Delta|\overline{b}|-\overline{b}^2/2\leq\Delta^2/2 Δ b sgn − b 2 /2 ≤ Δ∣ b ∣ − b 2 /2 ≤ Δ 2 /2
d P d Q ≤ exp ( γ T 2 + Δ 2 T 2 s 2 + U 0 ( v 0 ) − U 0 ( V 0 ( T ) ) s 2 ) . (32) \frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}\leq\exp\left(\frac{\gamma T}{2}+\frac{\Delta^2T}{2s^2}+\frac{U_0(v_0)-U_0(V^0(T))}{s^2}\right). \tag*{(32)} d Q d P ≤ exp ( 2 γ T + 2 s 2 Δ 2 T + s 2 U 0 ( v 0 ) − U 0 ( V 0 ( T )) ) . (32) Step 3 (law under Q \mathbb{Q} Q By (B.6) and Step 1,
P ( V 0 ( T ) ∈ E ) = E Q [ d P d Q 1 E ( V 0 ( T ) ) ] ≤ ∫ E e γ T 2 + Δ 2 T 2 s 2 + U 0 ( v 0 ) − U 0 ( v ) s 2 p T ( I ) ( v ) d v = ∫ E p ‾ T ( v ) d v , \mathbb{P}(V^0(T)\in E)=\mathbb{E}_{\mathbb{Q}}\left[\frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}}\mathbf{1}_E(V^0(T))\right]\leq\int_E e^{\frac{\gamma T}{2}+\frac{\Delta^2T}{2s^2}+\frac{U_0(v_0)-U_0(v)}{s^2}}p_T^{(I)}(v)\,\mathrm{d}v=\int_E\overline{p}_T(v)\,\mathrm{d}v, P ( V 0 ( T ) ∈ E ) = E Q [ d Q d P 1 E ( V 0 ( T )) ] ≤ ∫ E e 2 γ T + 2 s 2 Δ 2 T + s 2 U 0 ( v 0 ) − U 0 ( v ) p T ( I ) ( v ) d v = ∫ E p T ( v ) d v , where the last equality follows from (B.5), (B.2) and U = U 0 + Δ ∣ ⋅ ∣ U=U_0+\Delta|\cdot| U = U 0 + Δ∣ ⋅ ∣
Step 4 (consequences). By (B.4), P ( V 0 ( T ) ∈ E ) = 0 \mathbb{P}(V^0(T)\in E)=0 P ( V 0 ( T ) ∈ E ) = 0 Leb ( E ) = 0 \operatorname{Leb}(E)=0 Leb ( E ) = 0 V 0 ( T ) V^0(T) V 0 ( T ) p T p_T p T p T ≤ p ‾ T p_T\leq\overline{p}_T p T ≤ p T E = { p T > p ‾ T } E=\{p_T>\overline{p}_T\} E = { p T > p T } U ( v ) ≥ U ∗ U(v)\geq U_* U ( v ) ≥ U ∗ e − Δ ( ∣ v ∣ + ∣ v 0 ∣ ) / s 2 ≤ 1 e^{-\Delta(|v|+|v_0|)/s^2}\leq1 e − Δ ( ∣ v ∣ + ∣ v 0 ∣ ) / s 2 ≤ 1 Φ ≤ 1 \Phi\leq1 Φ ≤ 1 1 1 1 Leb ( E ) > 0 \operatorname{Leb}(E)>0 Leb ( E ) > 0 Q ( V 0 ( T ) ∈ E ) = ∫ E p T ( I ) ( v ) d v > 0 \mathbb{Q}(V^0(T)\in E)=\int_E p_T^{(I)}(v)\,\mathrm{d}v>0 Q ( V 0 ( T ) ∈ E ) = ∫ E p T ( I ) ( v ) d v > 0 d P / d Q > 0 \mathrm{d}\mathbb{P}/\mathrm{d}\mathbb{Q}>0 d P / d Q > 0 P ( V 0 ( T ) ∈ E ) > 0 \mathbb{P}(V^0(T)\in E)>0 P ( V 0 ( T ) ∈ E ) > 0 V 0 ( T ) V^0(T) V 0 ( T )
Algorithm 1 Coupled simulation of ( V ε ( T ) , V 0 ( T ) ) (V^\varepsilon(T), V^0(T)) ( V ε ( T ) , V 0 ( T ))
Require: ε \varepsilon ε T T T n n n δ t = T / n \delta t = T/n δ t = T / n A \mathbf{A} A K \mathbf{K} K r \mathbf{r} r σ \sigma σ Δ \Delta Δ γ \gamma γ ℓ \ell ℓ v 0 v_0 v 0
draw η ∼ N ( 0 , C ) \boldsymbol{\eta} \sim\mathcal{N}(0,\mathbf{C}) η ∼ N ( 0 , C ) V ε ← v 0 V^\varepsilon\leftarrow v_0 V ε ← v 0 V 0 ← v 0 V^0 \leftarrow v_0 V 0 ← v 0 c ← A − ⊤ r \mathbf{c} \leftarrow\mathbf{A}^{-\top}\mathbf{r} c ← A − ⊤ r h ← 0 h \leftarrow0 h ← 0 Z ∗ ← 0 Z^* \leftarrow0 Z ∗ ← 0 H ∗ ← 0 H^* \leftarrow0 H ∗ ← 0
for m = 0 , … , n − 1 m=0,\ldots,n-1 m = 0 , … , n − 1
draw ζ ∼ N ( 0 , I d ′ ) \boldsymbol{\zeta} \sim\mathcal{N}(0,\mathbf{I}_{d'}) ζ ∼ N ( 0 , I d ′ )
δ N ε ← σ δ t r ⋅ η / ε \delta N^\varepsilon\leftarrow\sigma\delta t\,\mathbf{r}\cdot\boldsymbol{\eta}/\varepsilon δ N ε ← σ δ t r ⋅ η / ε δ N 0 ← σ δ t c ⊤ K ζ \delta N^0 \leftarrow\sigma\sqrt{\delta t}\,\mathbf{c}^{\top}\mathbf{K}\boldsymbol{\zeta} δ N 0 ← σ δ t c ⊤ K ζ h ← h + δ N ε − δ N 0 h \leftarrow h+\delta N^\varepsilon-\delta N^0 h ← h + δ N ε − δ N 0
V ε ← soft Δ δ t ( V ε + ℓ δ t + δ N ε ) / ( 1 + γ δ t ) V^\varepsilon\leftarrow\operatorname{soft}_{\Delta\delta t}(V^\varepsilon+\ell\delta t+\delta N^\varepsilon)/(1+\gamma\delta t) V ε ← soft Δ δ t ( V ε + ℓ δ t + δ N ε ) / ( 1 + γ δ t )
V 0 ← soft Δ δ t ( V 0 + ℓ δ t + δ N 0 ) / ( 1 + γ δ t ) V^0 \leftarrow\operatorname{soft}_{\Delta\delta t}(V^0+\ell\delta t+\delta N^0)/(1+\gamma\delta t) V 0 ← soft Δ δ t ( V 0 + ℓ δ t + δ N 0 ) / ( 1 + γ δ t )
η ← η − ( δ t / ε 2 ) A η + ( δ t / ε ) K ζ \boldsymbol{\eta} \leftarrow\boldsymbol{\eta}-(\delta t/\varepsilon^2)\mathbf{A}\boldsymbol{\eta}+(\sqrt{\delta t}/\varepsilon)\mathbf{K}\boldsymbol{\zeta} η ← η − ( δ t / ε 2 ) A η + ( δ t / ε ) K ζ ζ \boldsymbol{\zeta} ζ
Z ∗ ← max { Z ∗ , ∣ V ε − V 0 ∣ } Z^* \leftarrow\max\{Z^*,|V^\varepsilon-V^0|\} Z ∗ ← max { Z ∗ , ∣ V ε − V 0 ∣ } H ∗ ← max { H ∗ , ∣ h ∣ } H^* \leftarrow\max\{H^*,|h|\} H ∗ ← max { H ∗ , ∣ h ∣ } Z ∗ ≤ 2 H ∗ Z^* \le2H^* Z ∗ ≤ 2 H ∗ Proposition 5.1
end for
return ( V ε , V 0 , Z ∗ , H ∗ ) (V^\varepsilon,V^0,Z^*,H^*) ( V ε , V 0 , Z ∗ , H ∗ )
Implementation details
Algorithm. Algorithm 1 summarizes the simulation of one coupled path with the backward Euler scheme (23) .
Semi-implicit variant. The update V m + 1 = soft Δ δ t ( ( 1 − γ δ t ) V m + ℓ δ t + δ N m ) V_{m+1}=\operatorname{soft}_{\Delta\delta t}((1-\gamma\delta t)V_m+\ell\delta t+\delta N_m) V m + 1 = soft Δ δ t (( 1 − γ δ t ) V m + ℓ δ t + δ N m ) V m + 1 = V m + δ N m + δ t ( b ˉ ( V m ) − k m + 1 ) V_{m+1}=V_m+\delta N_m+\delta t(\bar b(V_m)-k_{m+1}) V m + 1 = V m + δ N m + δ t ( b ˉ ( V m ) − k m + 1 ) k m + 1 ∈ ∂ ϕ ( V m + 1 ) k_{m+1}\in\partial\phi(V_{m+1}) k m + 1 ∈ ∂ ϕ ( V m + 1 ) γ δ t ≤ 1 \gamma\delta t\le1 γ δ t ≤ 1 Proposition 5.1 holds for it as well: with the notation of its proof, Λ m + 1 = ( 1 − γ δ t ) Λ m − γ δ t h m − δ t ( k m + 1 ε − k m + 1 0 ) \Lambda_{m+1}=(1-\gamma\delta t)\Lambda_m-\gamma\delta t\,h_m-\delta t(k^\varepsilon_{m+1}-k^0_{m+1}) Λ m + 1 = ( 1 − γ δ t ) Λ m − γ δ t h m − δ t ( k m + 1 ε − k m + 1 0 ) Λ m + 1 > H m + 1 \Lambda_{m+1}>H_{m+1} Λ m + 1 > H m + 1 k m + 1 ε ≥ k m + 1 0 k^\varepsilon_{m+1}\ge k^0_{m+1} k m + 1 ε ≥ k m + 1 0 Λ m + 1 ≤ ( 1 − γ δ t ) H m + γ δ t H m = H m \Lambda_{m+1}\le(1-\gamma\delta t)H_m+\gamma\delta t\,H_m=H_m Λ m + 1 ≤ ( 1 − γ δ t ) H m + γ δ t H m = H m § 5.4 are obtained with (23) .
Time step. We take δ t = T / n \delta t=T/n δ t = T / n n = ⌈ T / min { 10 − 3 , ε 2 / 50 } ⌉ n=\lceil T/\min\{10^{-3},\varepsilon^2/50\}\rceil n = ⌈ T / min { 1 0 − 3 , ε 2 /50 }⌉ A = 1 \mathbf{A}=1 A = 1 δ t A / ε 2 ≤ 1 / 50 \delta t A/\varepsilon^2\le1/50 δ t A / ε 2 ≤ 1/50 ε \varepsilon ε A \mathbf{A} A δ t ≤ ε 2 / ( 50 ∥ A ∥ ) \delta t\le\varepsilon^2/(50\|\mathbf{A}\|) δ t ≤ ε 2 / ( 50∥ A ∥ ) ε \varepsilon ε ε 2 / 50 ≤ 10 − 3 \varepsilon^2/50\le10^{-3} ε 2 /50 ≤ 1 0 − 3 1 − δ t A / ε 2 1-\delta t A/\varepsilon^2 1 − δ t A / ε 2 49 / 50 49/50 49/50 n n n K 2 / ( 2 A ) \mathbf{K}^2/(2A) K 2 / ( 2 A ) ( 1 − δ t A / ( 2 ε 2 ) ) − 1 ≤ 1.0102 (1-\delta t A/(2\varepsilon^2))^{-1}\le1.0102 ( 1 − δ t A / ( 2 ε 2 ) ) − 1 ≤ 1.0102 ε \varepsilon ε
Random numbers and standard errors. Batch B \mathcal{B} B ε \varepsilon ε [56 ] of NumPy, seeded by SeedSequence(42424242, spawn_key=(stage, regime, round(10^6\varepsilon), \mathcal{B})), so that all streams are independent and the results are reproducible. With s ( ⋅ ) s(\cdot) s ( ⋅ ) N N N s ( ( D ε ) 2 ) N − 1 / 2 s((D^\varepsilon)^2)N^{-1/2} s (( D ε ) 2 ) N − 1/2 D 2 D_2 D 2 s ( F ε ) N − 1 / 2 s(F^\varepsilon)N^{-1/2} s ( F ε ) N − 1/2 s ( D ε ) N − 1 / 2 s(D^\varepsilon)N^{-1/2} s ( D ε ) N − 1/2 σ ^ K N − 1 / 2 \hat{\sigma}_K N^{-1/2} σ ^ K N − 1/2 I ^ N \hat I_N I ^ N J ^ N ε \hat J_N^\varepsilon J ^ N ε K ^ N ε \hat K_N^\varepsilon K ^ N ε σ ^ K 2 = s ( F ε ) 2 − λ ^ N ε Cov ( F ε , F 0 ) \hat{\sigma}_K^2=s(F^\varepsilon)^2-\hat{\lambda}_N^\varepsilon\operatorname{Cov}(F^\varepsilon,F^0) σ ^ K 2 = s ( F ε ) 2 − λ ^ N ε Cov ( F ε , F 0 ) s ( D ε ) N − 1 / 2 s(D^\varepsilon)N^{-1/2} s ( D ε ) N − 1/2 E [ D ε ] \mathbb{E}[D^\varepsilon] E [ D ε ] λ ^ N ε \hat{\lambda}_N^\varepsilon λ ^ N ε N N N s B / B s_{\mathcal{B}}/\sqrt{\mathcal{B}} s B / B s B s_{\mathcal{B}} s B B = 12 \mathcal{B}=12 B = 12
Local slopes and fitted exponents. For consecutive grid values ε i > ε i + 1 \varepsilon_i>\varepsilon_{i+1} ε i > ε i + 1
r i ≜ log D 2 ( ε i + 1 ) − log D 2 ( ε i ) log ε i + 1 − log ε i r_i \triangleq\frac{\log D_2(\varepsilon_{i+1})-\log D_2(\varepsilon_i)}{\log\varepsilon_{i+1}-\log\varepsilon_i} r i ≜ log ε i + 1 − log ε i log D 2 ( ε i + 1 ) − log D 2 ( ε i ) is plotted at ε ˉ i ≜ ( ε i ε i + 1 ) 1 / 2 \bar{\varepsilon}_i\triangleq(\varepsilon_i\varepsilon_{i+1})^{1/2} ε ˉ i ≜ ( ε i ε i + 1 ) 1/2 ( e i 2 + e i + 1 2 ) 1 / 2 / ∣ log ( ε i + 1 / ε i ) ∣ (e_i^2+e_{i+1}^2)^{1/2}/|\log(\varepsilon_{i+1}/\varepsilon_i)| ( e i 2 + e i + 1 2 ) 1/2 /∣ log ( ε i + 1 / ε i ) ∣ e i e_i e i D 2 ( ε i ) D_2(\varepsilon_i) D 2 ( ε i ) log D 2 ( ε i ) \log D_2(\varepsilon_i) log D 2 ( ε i ) log ε i \log\varepsilon_i log ε i k k k ε i ≤ 0.05 \varepsilon_i\le0.05 ε i ≤ 0.05 e i − 2 e_i^{-2} e i − 2 ( ∑ i e i − 2 ( log ε i − log ε ‾ ) 2 ) − 1 / 2 (\sum_i e_i^{-2}(\log\varepsilon_i-\overline{\log\varepsilon})^2)^{-1/2} ( ∑ i e i − 2 ( log ε i − log ε ) 2 ) − 1/2 log ε ‾ \overline{\log\varepsilon} log ε ε i \varepsilon_i ε i ( χ 2 / ( k − 2 ) ) 1 / 2 (\chi^2/(k-2))^{1/2} ( χ 2 / ( k − 2 ) ) 1/2 χ 2 \chi^2 χ 2
Reference values. By Itô’s formula and Lemma A.1 , the laws of V 0 ( t ) V^0(t) V 0 ( t ) t ∈ [ 0 , T ] t \in[0,T] t ∈ [ 0 , T ]
∂ t p = ∂ v [ ( γ v + Δ sgn ( v ) − ℓ ) p ] + s 2 2 ∂ v 2 p , p ( 0 , ⋅ ) = δ v 0 . (33) \partial_t p = \partial_v\left[(\gamma v+\Delta\operatorname{sgn}(v)-\ell)p\right]+\frac{s^2}{2}\partial_v^2p,\qquad p(0,\cdot)=\delta_{v_0}. \tag*{(33)} ∂ t p = ∂ v [ ( γ v + Δ sgn ( v ) − ℓ ) p ] + 2 s 2 ∂ v 2 p , p ( 0 , ⋅ ) = δ v 0 . (33) which is [12 (1.4)] with noise power s 2 s^2 s 2 ℓ \ell ℓ p T p_T p T [12 (2.10)] , that is, by (B.5). In regimes (II)–(III), it is evaluated from the spectral series [12 (3.12), (4.7)] , truncated at 40 modes and converted to physical variables.
The eigenvalues are bracketed on a grid of step 0.05 and refined by Brent’s method. The parabolic cylinder functions D ν D_\nu D ν [57 ] ; since the library routine loses accuracy for orders slightly above a positive integer, D ν D_\nu D ν ν \nu ν ν \nu ν 10 − 4 10^{-4} 1 0 − 4
Since the completeness of these eigenfunction expansions is not proved in [12 ] , the truncated series are cross-validated against three independent computations: (i) the numerical inversion, by Talbot’s method [58 ] as in [13 Section 2] , of the closed-form Laplace transform [13 (7)–(9)] , evaluated with mpmath in 30-digit arithmetic at the test points ν ∈ { − 2 , − 0.5 , 0.25 , 1 , 2.5 } \nu\in\{-2,-0.5,0.25,1,2.5\} ν ∈ { − 2 , − 0.5 , 0.25 , 1 , 2.5 } [ − 20 , 20 ] [-20,20] [ − 20 , 20 ] [59 ] , zero-flux boundary conditions, and implicit Euler steps 4 ⋅ 10 − 4 4\cdot10^{-4} 4 ⋅ 1 0 − 4 2 ⋅ 10 − 4 2\cdot10^{-4} 2 ⋅ 1 0 − 4 f ( V n 0 ) f(V^0_n) f ( V n 0 ) ( 0 , ± v f (0,\pm v_f ( 0 , ± v f ± ( v f + 1 ) \pm(v_f+1) ± ( v f + 1 ) Table 3 reports the resulting reference values, the discrepancies, and sup p T \sup p_T sup p T p T ≤ p ˉ T p_T\le\bar p_T p T ≤ p ˉ T Proposition B.3 was also checked on a grid of [ − 10 , 10 ] [-10,10] [ − 10 , 10 ]
