On stochastic solutions of nonlocal random functional integral equations

In this paper, we use Schauder’s fixed point to establish the existence of at least one solution for a functional nonlocal stochastic differential equation under sufficient conditions in the space of all square integrable stochastic processes with a finite second moment. We state and prove the conditions which guarantee the uniqueness of the solution. We solve a nonlinear example analytically and obtain the initial condition which makes the solution passes through a random position with a given normal distribution at a specified time. Also, the Milstein scheme to this example is studied.