On a stochastic differential equation with correction term governed by a monotone and Lipschitz continuous operator

In our pursuit of finding a zero for a monotone and Lipschitz continuous operator $ M : \mathbb{R}^n \rightarrow \mathbb{R}^n $ amidst noisy evaluations, we explore an associated differential equation within a stochastic framework, incorporating a correction term. We present a result establishing the existence and uniqueness of solutions for the stochastic differential equations under examination. Additionally, assuming that the diffusion term is square-integrable, we demonstrate the almost sure convergence of the trajectory process $ X(t) $ to a zero of $ M $ and of $ \|M(X(t))\| $ to $ 0 $ as $ t \rightarrow +\infty $. Furthermore, we provide ergodic upper bounds and ergodic convergence rates in expectation for $ \|M(X(t))\|^2 $ and $ \langle M(X(t), X(t)-x^*\rangle $, where $ x^* $ is an arbitrary zero of the monotone operator. Subsequently, we apply these findings to a minimax problem. Finally, we analyze two temporal discretizations of the continuous-time models, resulting in stochastic variants of the Optimistic Gradient Descent Ascent and Extragradient methods, respectively, and assess their convergence properties.