Total variation distance between a jump-equation and its Gaussian approximation

TL;DR

This paper analyzes the accuracy of Gaussian approximations for jump-diffusion stochastic differential equations by quantifying the total variation distance between the original process and its approximation, providing convergence and density estimates.

Contribution

It proves convergence in total variation distance of the Gaussian-approximated process to the original jump process and estimates the approximation error using Malliavin calculus techniques.

Findings

Convergence of the approximation in total variation distance.

Error bounds for the approximation scheme.

Density difference estimates between the two processes.

Abstract

We deal with stochastic differential equations with jumps. In order to obtain an accurate approximation scheme, it is usual to replace the "small jumps" by a Brownian motion. In this paper, we prove that for every fixed time $t$, the approximate random variable $X^\varepsilon\_t$ converges to the original random variable $X\_t$ in total variation distance and we estimate the error. We also give an estimate of the distance between the densities of the laws of the two random variables. These are done by using some integration by parts techniques in Malliavin calculus.