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: ℝ ⁿ → 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 → +∞. Furthermore, we provide ergodic upper bounds and ergodic convergence rates in expectation for ∥M(X(t))∥ ² and ⟨M(X(t), X(t) − x ^∗⟩, 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.