Hyers-Ulam stability and continuous dependence of the solution of a nonlocal stochastic-integral problem of an arbitrary (fractional) orders stochastic differential equation

Abstract

Stochastic problems play a huge role in many applications including biology, chemistry, physics, economics, finance, mechanics, and several areas. In this paper, we are concerned with the nonlocal stochastic-integral problem of the arbitrary (fractional) orders stochastic differential equationwhere B is any Brownian motion, W is a standard Brownian motion, and X 0 is a second order random variable. The Hyers -Ulam stability of the problem will be studied. The existence of solution and its continuous dependence on the Brownian motion B will be proved. The three spatial cases Brownian bridge process, the Brownian motion with drift and the Brownian motion started at A will be considered.