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 $\beta$ 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 Lévy process has a Gaussian component, we obtain bounds of order ${\tau }^{-\beta /(6\beta -4)}$. In particular, the error is at most of order ${\tau }^{-1/6}$.

Published in at http://dx.doi.org/10.1214/10-AAP695 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• The Euler scheme for Levy driven stochastic differential equations: limit theorems

Cited by in corpus (21)

• Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations
• Multilevel Richardson-Romberg extrapolation
• Central limit theorem for the multilevel Monte Carlo Euler method
• Multilevel Monte Carlo for Lévy-driven SDEs: Central limit theorems for adaptive Euler schemes
• On the Lq(Lp)-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains
• Geometrically Convergent Simulation of the Extrema of Lévy Processes
• Pathwise approximation of SDEs by coupling piecewise abelian rough paths
• Discrepancy estimates for variance bounding Markov chain quasi-Monte Carlo
• Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation
• Implementable coupling of Lévy process and Brownian motion
• On the Acceleration of the Multi-Level Monte Carlo Method
• Improved adaptive Multilevel Monte Carlo and applications to finance
• A solution selection problem with small symmetric stable perturbations
• Multilevel Monte Carlo method for jump-diffusion SDEs
• Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation
• Multilevel approximation of backward stochastic differential equations
• Multilevel Monte Carlo For Exponential Lévy Models
• Multi-level stochastic approximation algorithms
• Multilevel Monte Carlo simulation for Levy processes based on the Wiener-Hopf factorisation
• Thinning and Multilevel Monte Carlo for Piecewise Deterministic (Markov) Processes. Application to a stochastic Morris-Lecar model
• A weak MLMC scheme for Lévy-copula-driven SDEs with applications to the pricing of credit, equity and interest rate derivatives