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 : ^n →^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.