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

Vlad Bally (1) , Yifeng Qin (1)

1 MATHRISK - Mathematical Risk Handling

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.

Mots clés: Total variation distance; Malliavin calculus; Gaussian approximation; Stochastic differential equations with jumps

Domaines: Probabilités [math.PR]

Vlad Bally, Yifeng Qin. Total variation distance between a jump-equation and its Gaussian approximation. Stochastics and Partial Differential Equations: Analysis and Computations, 2022, ⟨10.1007/s40072-022-00270-w⟩. ⟨hal-03351643v3⟩