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+\sigma(X_t) d B_t, \\ & Y_{t_{n+1}}=Y_{t_{n}}+\eta_{n+1} b(Y_{t_{n}})+\sigma(Y_{t_{n}})\left(B_{t_{n+1}}-B_{t_{n}}\right), \end{align*} where $b:\mathbb{R}^d \rightarrow \mathbb{R}^d,\ \ \sigma: \mathbb R^d \rightarrow \mathbb{R}^{d \times d}$ 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 $\lim_{k \rightarrow \infty} \eta_k=0$ and $\sum_{k=1}^\infty\eta_k =\infty$. Under (partial) dissipation conditions ensuring the ergodicity, we obtain explicit convergence rates of $\mathbb W_p(\mathscr{L}(Y_{t_n}), \mathscr{L}(X_{t_n}))+\mathbb W_p(\mathscr{L}(Y_{t_n}), \mu)\rightarrow 0$ as $n\rightarrow \infty$, where $\mathbb W_p$ is the $L^p$-Wasserstein distance for certain $p\in [0,\infty)$, $\mathscr{L}(\xi)$ 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.