Global Conference on Mathematics and Applications (GCMA)

We consider It\^o differential equations whose free evolution alternates with a prescribed impulse acting over an interval of positive duration. The objective is to quantify when these interventions produce a periodic probability law despite expansion during the free phase. The analysis combines the phase-composition estimates used for deterministic non-instantaneous impulsive equations with the pullback construction of random periodic solutions. Under global Lipschitz assumptions, a contraction of the complete-period map yields a unique adapted square-integrable random periodic solution, geometric convergence in the quadratic Wasserstein distance, and an explicit truncation bound. We also obtain an additive-noise perturbation estimate and a sufficient multiplicative-noise budget. For a scalar equation with independent additive and multiplicative Brownian noises, the budget is necessary and sufficient for a periodic law with finite second moment. An Euler--Maruyama approximation with exact impulse evaluation preserves the construction for sufficiently small steps. Exact moment calculations, checked independently by numerical integration, illustrate the noise threshold and the discretization bias. The results distinguish stabilization at period boundaries from fluctuations during the impulse interval.\[0.12cm]
\textbf{Keywords:} non-instantaneous impulses; random periodic solution; cycle contraction; periodic law; Euler--Maruyama approximation.\
\textbf{2020 Mathematics Subject Classification:} 34A37, 60H10, 37H05, 60H35.}