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

2020/12/28 by Gilles Pagès, Pages, Gilles, Fabien Panloup +1 · 7 citations
Economics, Econometrics and Finance · Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Statistical Methods and Inference #Statistics Theory (math.ST) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2012.14310

openalex publication_date 2020/12/28 · openalex created_date 2021/01/05 · openalex updated_date 2026/07/28

Abstract

In this paper, we focus on non-asymptotic bounds related to the Euler scheme\nof an ergodic diffusion with a possibly multiplicative diffusion term\n(non-constant diffusion coefficient). More precisely, the objective of this\npaper is to control the distance of the standard Euler scheme with decreasing\nstep (usually called Unadjusted Langevin Algorithm in the Monte Carlo\nliterature) to the invariant distribution of such an ergodic diffusion. In an\nappropriate Lyapunov setting and under uniform ellipticity assumptions on the\ndiffusion coefficient, we establish (or improve) such bounds for Total\nVariation and L¹-Wasserstein distances in both multiplicative and additive\nand frameworks. These bounds rely on weak error expansions using Stochastic\nAnalysis adapted to decreasing step setting.\n

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