Unadjusted Langevin Algorithms for SDEs with Hoelder Drift

Abstract

Consider the following stochastic differential equation for $(X_t{)}_{t\ge 0}$ on ${\mathbb{R}}^d$ and its Euler-Maruyama (EM) approximation $(Y_{t_n}{)}_{n\in {\mathbb{Z}}^{+}}$: \begin{align*} &d X_t=b( X_t) d t+σ(X_t) d B_t, \\ & Y_{t_{n+1}}=Y_{t_{n}}+η_{n+1} b(Y_{t_{n}})+σ(Y_{t_{n}})\left(B_{t_{n+1}}-B_{t_{n}}\right), \end{align*} where $b:{\mathbb{R}}^d\to {\mathbb{R}}^d,\sigma :{\mathbb{R}}^d\to {\mathbb{R}}^{d\times d}$ are measurable, $B_t$ is the $d$-dimensional Brownian motion, $t_0:=0,t_n:={\sum }_{k=1}^n{\eta }_k$ for constants ${\eta }_k>0$ satisfying ${\lim }_{k\to \infty }{\eta }_k=0$ and ${\sum }_{k=1}^{\infty }{\eta }_k=\infty$. Under (partial) dissipation conditions ensuring the ergodicity, we obtain explicit convergence rates of ${\mathbb{W}}_p(\mathcal{L}(Y_{t_n}),\mathcal{L}(X_{t_n}))+{\mathbb{W}}_p(\mathcal{L}(Y_{t_n}),\mu )\to 0$ as $n\to \infty$, where ${\mathbb{W}}_p$ is the $L^p$-Wasserstein distance for certain $p\in [0,\infty )$, $\mathcal{L}(\xi )$ is the distribution of random variable $\xi$, and $\mu$ is the unique invariant probability measure of $(X_t{)}_{t\ge 0}$. Comparing with the existing results where $b$ is at least $C^2$-smooth, our estimates apply to Hoelder continuous drift and can be sharp in several specific situations.