WELL-POSEDNESS AND REGULARITY OF SOLUTIONS FOR CAPUTO-HADAMARD STOCHASTIC FRACTIONAL DIFFERENTIAL EQUATIONS

Abstract

This paper first investigates the well-posedness and regularity of solutions to Caputo–Hadamard stochastic fractional differential equations (CHSFDEs) of order λ ∈ ( 1/2, 1) in Lp spaces with p ≥ 2, driven by a Brownian motion under global Lipschitz conditions on the coefficients. In particular, the existence and uniqueness of solutions, as well as their continuous dependence on both the initial data and the fractional order, are proved. Furthermore, the regularity of solutions is investigated. As a direct consequence, an upper bound for the moment of the solution in the supremum norm is obtained via the application of the Garsia–Rodemich–Rumsey (GRR) lemma. The analysis relies on a weighted norm together with the Banach fixed-point theorem, the H¨older’s inequality, the Burkholder–Davis–Gundy (BDG) inequality, generalized Gr¨onwall inequality for Caputo–Hadamard fractional differential equations, properties of the logarithmic kernel, and GRR lemma.