Analysis and discretization of nonlinear generalized fractional stochastic differential equations

Abstract

This paper investigates the well-posedness and numerical approximation of a nonlinear generalized fractional stochastic differential equation driven by multiplicative white noise. The proposed model generalizes conventional Caputo-type fractional stochastic systems by incorporating a kernel function $\zeta \left(t\right)$, which enables flexible characterization of memory effects and nonlocal interactions in complex stochastic dynamics. The existence, uniqueness, and regularity of solutions for the considered problem is established by combining generalized Gronwall inequality with smoothness assumptions on $\zeta \left(t\right)$. Furthermore, a novel generalized Euler–Maruyama scheme is constructed for numerical resolution, whose stability and strong convergence (under refined regularity conditions on $\zeta \left(t\right)$) are rigorously proved. Compared with existing works limited to specific fractional operators, our methodology provides a unified treatment for diverse kernel-driven stochastic systems while preserving numerical tractability. Several numerical experiments are included to verify the theoretical results. This work extends the modeling capability of fractional stochastic calculus and offers a versatile tool for describing anomalous diffusion processes and memory-dependent phenomena in interdisciplinary applications.