Wasserstein distance estimates for jump-diffusion processes

We derive Wasserstein distance bounds between the probability distributions of a stochastic integral (Itô) process with jumps $(X_t{)}_{t\in [0,T]}$ and a jump-diffusion process $(X_t^{\ast }{)}_{t\in [0,T]}$. Our bounds are expressed using the stochastic characteristics of $(X_t{)}_{t\in [0,T]}$ and the jump-diffusion coefficients of $(X_t^{\ast }{)}_{t\in [0,T]}$ 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.