Variation of Constants for Riemann--Liouville Fractional Stochastic Differential Equations with Sharp $L^2$-Threshold and Optimal Regularity

Abstract

We develop a rigorous analytical framework for Riemann-Liouville fractional  stochastic differential equations (RL-FSDEs), addressing a gap in the literature where existing treatments either focus on Caputo formulations or  provide incomplete existence arguments. Our approach combines Laplace transform operational calculus with It\^{o} stochastic integration theory to derive explicit variation-of-constants representations in terms of Mittag-Leffler convolution kernels. We establish that the threshold $\alpha > \tfrac{1}{2}$ is both necessary and sufficient for $L^2(\Omega)$-well-posedness of the Riemann-Liouville--It\^{o} stochastic integral, proved from first principles via a complete kernel integrability analysis. Under standard Lipschitz and linear growth conditions, we prove existence and uniqueness of mild solutions in both the pathwise and strong senses, with sharp moment bounds obtained through a Banach fixed-point argument in a weighted $L^p$ space. Optimal H\"{o}lder regularity of exponent $(\alpha - \tfrac{1}{2} - \varepsilon)$ is established via the Kolmogorov--Chentsov criterion, and is shown to be unimprovable. We further verify rigorously that mild solutions satisfy the prescribed Riemann-Liouville initial conditions and address the progressive measurability of kernels depending on the terminal time. The theory is illustrated through four detailed examples---including fractional Ornstein--Uhlenbeck processes, coupled two-dimensional systems, time-varying drift, and harmonically forced equations---each with explicit closed-form solutions, moment functions, and asymptotic stability analysis. This self-contained treatment provides a foundation for numerical methods, statistical inference, and optimal control of RL-FSDEs.