An operator approach for Markov chain weak approximations with an application to infinite activity L\'{e}vy driven SDEs

TL;DR

This paper introduces a semigroup expansion framework for constructing higher-order weak approximation schemes for stochastic differential equations, including those driven by Lévy processes, and analyzes their convergence rates.

Contribution

It develops a general operator-based framework for higher-order weak approximations and applies it to Lévy-driven SDEs, extending existing numerical methods.

Findings

Framework achieves higher-order convergence rates.

Applicable to Lévy-driven SDEs with general jump structures.

Provides theoretical analysis of convergence rates.

Abstract

Weak approximations have been developed to calculate the expectation value of functionals of stochastic differential equations, and various numerical discretization schemes (Euler, Milshtein) have been studied by many authors. We present a general framework based on semigroup expansions for the construction of higher-order discretization schemes and analyze its rate of convergence. We also apply it to approximate general L\'{e}vy driven stochastic differential equations.