Fractional neutral stochastic differential equations with Caputo fractional derivative: Fractional Brownian motion, Poisson jumps, and optimal control

Abstract

The objective of this paper is to investigate the existence of mild solutions and optimal controls for a class of fractional neutral stochastic differential equations (NSDEs) driven by fractional Brownian motion and Poisson jumps in Hilbert spaces. First, we establish a new set of sufficient conditions for the existence of mild solutions of the aforementioned fractional systems by using the successive approximation approach. The results are formulated and proved by using the fractional calculus, solution operator, and stochastic analysis techniques. The existence of optimal control pairs of system governed by fractional NSDEs driven by fractional Brownian motion and Poisson jumps is also been presented. An example is provided to illustrate the theory.