Ulam stability and continuous dependence of the solution of a nonlocal stochastic-integral fractional orders stochastic differential equation

Abstract

Stochastic problems have become an indispensable tool in modeling complex systems across various disciplines, including biology, chemistry, physics, economics, finance, mechanics and several areas. In this paper, we are concerning with the nonlocal problem of the integro-fractional orders stochastic differential equation dX(t) dt = f(t,DX(t)) + g(t,B(t)), t (0, T], with the nonlocal stochastic-integral conditiom X() = X0 + Z T− 0 h(s,DX(s))dW(s),[0, T] where W is a standard Brownian motion, B is any Brownian motion and X0 is a second order random variable. The existence of solution and its continuous dependencies on X0, the functions f(t, x), g(t, x) and on the Brownian motion B will be discussed. Finally the Hyers - Ulam stability of the problem will be studied.