Unadjusted Langevin Algorithms for SDEs with Hoelder Drift

2023/09/30 by Li, Xiang, Wang, Feng-Yu, Xu, Lihu · 3 citations
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2310.00232

Abstract

Consider the following stochastic differential equation for (X_t)_{t≥ 0} on \mathbb R^d and its Euler-Maruyama (EM) approximation (Y_tn)_{n∈ \mathbb Z⁺}: amp;d X_t=b( X_t) d t+σ(X_t) d B_t,
amp; Y_t_n+1=Y_t_n+η_n+1 b(Y_t_n)+σ(Y_t_n)(B_t_n+1-B_t_n), where b:ℝ^d → ℝ^d, σ: \mathbb R^d → ℝ^{d × d} are measurable, B_t is the d-dimensional Brownian motion, t₀:=0,t_n:=∑_k=1ⁿ η_k for constants η_k>0 satisfying lim_{k → ∞} η_k=0 and ∑_k=1^∞η_k =∞. Under (partial) dissipation conditions ensuring the ergodicity, we obtain explicit convergence rates of \mathbb W_p(\mathscrL(Y_tn), \mathscrL(X_tn))+\mathbb W_p(\mathscrL(Y_tn), μ)→ 0 as n→ ∞, where \mathbb W_p is the L^p-Wasserstein distance for certain p∈ [0,∞), \mathscrL(ξ) is the distribution of random variable ξ, and μ is the unique invariant probability measure of (X_t)_{t ≥ 0}. Comparing with the existing results where b is at least C²-smooth, our estimates apply to Hoelder continuous drift and can be sharp in several specific situations.

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