Multilevel Monte Carlo algorithms for L\'{e}vy-driven SDEs with Gaussian correction

TL;DR

This paper develops multilevel Monte Carlo algorithms for efficiently computing expectations of functionals of solutions to Lévy-driven SDEs, utilizing Gaussian approximations for small jumps and analyzing error bounds based on Lévy measure properties.

Contribution

It introduces a novel multilevel Monte Carlo method combining jump simulation with Gaussian approximation, providing explicit error bounds linked to the Lévy measure's behavior.

Findings

Error bounds depend on the Blumenthal–Getoor index of the Lévy process.

The algorithm outperforms non-Gaussian approaches when the Blumenthal–Getoor index exceeds one.

Error order varies with the Lévy process's characteristics, achieving at most 1/6 order.

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 L\'{e}vy-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 L\'{e}vy 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 L\'{e}vy measure around zero. In particular, one can derive upper bounds from the Blumenthal--Getoor index of the L\'{e}vy 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 L\'{e}vy process does not incorporate a Wiener process or if the Blumenthal--Getoor index $\beta$ is larger than $\frac{4}{3}$, then the upper bound is of order $\tau^{-({4-\beta})/({6\beta})}$ when the runtime $\tau$ tends to infinity. Whereas in the case, where $\beta$ is in $[1,\frac{4}{3}]$ and the L\'{e}vy process has a Gaussian component, we obtain bounds of order $\tau^{-\beta/(6\beta-4)}$. In particular, the error is at most of order $\tau^{-1/6}$.