Algebraic Structures and Stochastic Differential Equations driven by Levy processes

TL;DR

This paper develops an efficient, highly accurate integrator for stochastic differential equations driven by Levy processes, leveraging algebraic structures to achieve optimal convergence properties.

Contribution

It introduces a novel integrator that surpasses stochastic Taylor methods in accuracy and is optimal within a broad class, using quasi-shuffle algebra of Levy process integrals.

Findings

The integrator is more accurate than stochastic Taylor approximations to all orders.

It is optimal within a broad class of perturbations for certain convergence orders.

The method applies to Levy-driven systems with moments of all orders and smooth vector fields.

Abstract

We construct an efficient integrator for stochastic differential systems driven by Levy processes. An efficient integrator is a strong approximation that is more accurate than the corresponding stochastic Taylor approximation, to all orders and independent of the governing vector fields. This holds provided the driving processes possess moments of all orders and the vector fields are sufficiently smooth. Moreover the efficient integrator in question is optimal within a broad class of perturbations for half-integer global root mean-square orders of convergence. We obtain these results using the quasi-shuffle algebra of multiple iterated integrals of independent Levy processes.