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

TL;DR

This paper proves that approximating jump-diffusion processes with Gaussian processes converges in total variation distance, providing error estimates and density comparisons using Malliavin calculus techniques.

Contribution

It establishes convergence and error bounds for Gaussian approximations of jump processes in total variation distance, with density estimates.

Findings

Convergence of the approximation in total variation distance.

Error estimates for the approximation.

Density law distance bounds.

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.