Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds

In this paper, we focus on nonasymptotic bounds related to the Euler scheme of an ergodic diffusion with a possibly multiplicative diffusion term (nonconstant diffusion coefficient). More precisely, the objective of this paper is to control the distance of the standard Euler scheme with decreasing step (usually called unadjusted Langevin algorithm in the Monte Carlo literature) to the invariant distribution of such an ergodic diffusion. In an appropriate Lyapunov setting and under uniform ellipticity assumptions on the diffusion coefficient, we establish (or improve) such bounds for total variation and L1-Wasserstein distances in both multiplicative and additive and frameworks. These bounds rely on weak error expansions using stochastic analysis adapted to decreasing step setting.