Weak Error for stable driven SDEs: expansion of the densities

TL;DR

This paper derives a first-order error expansion for the density difference between a stable-driven SDE and its Euler approximation, using a parametrix method under certain coefficient assumptions.

Contribution

It introduces a novel error expansion for densities of stable-driven SDEs and their Euler schemes, advancing numerical analysis in this context.

Findings

First-order density error expansion derived

Applicable to multidimensional stable-driven SDEs

Provides theoretical foundation for numerical schemes

Abstract

Consider a multidimensional SDE of the form $X_t = x+\int_{0}^{t} b(X_{s-})ds+\int{0}^{t} f(X_{s-})dZ_s$ where $(Z_s)_{s\ge 0}$ is a symmetric stable process. Under suitable assumptions on the coefficients the unique strong solution of the above equation admits a density w.r.t. the Lebesgue measure and so does its Euler scheme. Using a parametrix approach, we derive an error expansion at order 1 w.r.t. the time step for the difference of these densities.