Wasserstein distance estimates for jump-diffusion processes

TL;DR

This paper establishes bounds on the Wasserstein distance between distributions of jump-diffusion processes and stochastic integrals, using stochastic calculus and $L^p$ integrability, applicable even with differing jump characteristics.

Contribution

It provides a novel method to estimate Wasserstein distances between jump-diffusion processes without relying on Stein equations, incorporating stochastic calculus techniques.

Findings

Derived explicit Wasserstein bounds for jump-diffusion processes

Applicable to processes with different jump characteristics

Utilizes stochastic calculus and $L^p$ integrability methods

Abstract

We derive Wasserstein distance bounds between the probability distributions of a stochastic integral (It\^o) process with jumps $(X_t)_{t\in [0,T]}$ and a jump-diffusion process $(X^\ast_t)_{t\in [0,T]}$. Our bounds are expressed using the stochastic characteristics of $(X_t)_{t\in [0,T]}$ and the jump-diffusion coefficients of $(X^\ast_t)_{t\in [0,T]}$ evaluated in $X_t$, and apply in particular to the case of different jump characteristics. Our approach uses stochastic calculus arguments and $L^p$ integrability results for the flow of stochastic differential equations with jumps, without relying on the Stein equation.