Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction

Abstract

We introduce and analyze multilevel Monte Carlo algorithms for the computation of $\mathbb {E}f(Y)$, where $Y=(Y_t)_{t\in[0,1]}$ is the solution of a multidimensional Lévy-driven stochastic differential equation and $f$ is a real-valued function on the path space. The algorithm relies on approximations obtained by simulating large jumps of the Lévy process individually and applying a Gaussian approximation for the small jump part. Upper bounds are provided for the worst case error over the class of all measurable real functions $f$ that are Lipschitz continuous with respect to the supremum norm. These upper bounds are easily tractable once one knows the behavior of the Lévy measure around zero. In particular, one can derive upper bounds from the Blumenthal–Getoor index of the Lévy process. In the case where the Blumenthal–Getoor index is larger than one, this approach is superior to algorithms that do not apply a Gaussian approximation. If the Lévy process does not incorporate a Wiener process or if the Blumenthal–Getoor index $β$ is larger than $\frac{4}{3}$, then the upper bound is of order $τ^{-({4-β})/({6β})}$ when the runtime $τ$ tends to infinity. Whereas in the case, where $β$ is in $[1,\frac{4}{3}]$ and the Lévy process has a Gaussian component, we obtain bounds of order $τ^{-β/(6β-4)}$. In particular, the error is at most of order $τ^{-1/6}$.