An Euler-Poisson Scheme for L\'evy driven SDEs

TL;DR

This paper introduces an Euler scheme with random Poisson grid points for approximating solutions to Le9vy driven SDEs, providing convergence analysis and applicability to various Le9vy processes.

Contribution

It extends previous work by analyzing a new Euler scheme on random Poisson grids for Le9vy SDEs with proven convergence rates and broad applicability.

Findings

Convergence rate of a0b1 n^{-1/2} in mean square error.

Applicability to stable, spectrally one-sided, and meromorphic Le9vy processes.

Implementation simplicity for processes with known resolvent samples.

Abstract

We describe an Euler scheme to approximate solutions of L\'evy driven Stochastic Differential Equations (SDE) where the grid points are random and given by the arrival times of a Poisson process. This result extends a previous work of the authors in Ferreiro-Castilla et al. (2012). We provide a complete numerical analysis of the algorithm to approximate the terminal value of the SDE and proof that the approximation converges in mean square error with rate $\mathcal{O}(n^{-1/2})$. The only requirement of the methodology is to have exact samples from the resolvent of the L\'evy process driving the SDE; classic examples such as stable processes, subclasses of spectrally one sided L\'evy processes and new families such as meromorphic L\'evy processes (cf. Kuznetsov et al. (2011)) are some examples for which the implementation of our algorithm is straightforward.