A variation of constant formula for Caputo–Hadamard fractional stochastic differential equations⋆

Abstract

This paper studies the existence and uniqueness of the mild solutions of Caputo–Hadamard fractional stochastic differential equations (SDEs). Subsequently, a variation of constants formula is derived for these equations. The primary proof techniques rely on Itô’s isometry, the martingale representation theorem, and the adaptation of the variation of constants formula employed in deterministic Caputo–Hadamard fractional differential equations (FDEs). Furthermore, we employ the constant variation formula to investigate the mean-square stability of a class of scalar Caputo–Hadamard fractional SDEs and provide stability criteria. Consequently, this class of scalar equations can serve as basic test equations to study the stability of numerical methods for Caputo–Hadamard fractional SDEs.