Weak Error on the densities for the Euler scheme of stable additive SDEs with Hölder drift

Abstract

We are interested in the Euler–Maruyama discretization of the SDE $d{X}_t=b(t,X_t)dt+d{Z}_t,X_0=x\in R^d,$where $Z_t$ is a symmetric isotropic $d$-dimensional $\alpha$-stable process, $\alpha \in (1,2]$ and the drift $b\in L^{\infty }\left([0,T],C^{\beta }(R^d,R^d)\right)$, $\beta \in (0,1)$, is bounded and Hölder regular in space. Using an Euler scheme with a randomization of the time variable, we show that, denoting $\gamma ≔\alpha +\beta -1$, the weak error on densities related to this discretization converges at the rate $\gamma /\alpha$.