Well-posedness and regularity for solutions of Caputo stochastic fractional delay differential equations

Abstract

This paper is devoted to build the well-posedness and regularity for solutions of Caputo stochastic fractional delay differential equations (for short CSFDDE) of order α∈(12,1). Firstly, under local Lipschitz condition of coefficients, we show a result on the existence and uniqueness of solutions. Secondly, under global Lipschitz condition of coefficients, we show the continuous dependence of solutions on the initial values and on the fractional exponent α and the regularity in time for solutions is also derived. The main ingredient in the proof is to use a temporally weighted norm, Banach fixed point theorem and truncation procedure.