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In this paper, we consider Caputo type fractional stochastic time-delay system with permutable matrices. We derive stochastic analogue of variation of constants formula via a newly defined delayed Mittag-Leffer type matrix function. Thus, we investigate new results on existence and uniqueness of mild solutions with the help of weighted maximum norm to fractional stochastic time-delay differential equations whose coefficients satisfy standard Lipschitz conditions. The main points in the proof are to apply Ito's isometry and martingale representation theorem, and to show the notion of a coincidence between the integral equation and the mild solution. Finally, we study complete controllability results for linear and nonlinear fractional stochastic delay dynamical systems with Wiener noise.

Controllability of fractional stochastic delay dynamical systems