  In our pursuit of finding a zero for a monotone and Lipschitz continuous
operator $M : \R^n \rightarrow \R^n$ 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.
