Almost automorphic solutions for fractional stochastic differential equations and its optimal control

Fractional calculus is the field of mathematical analysis that deals with the investigation and applications of integrals, derivatives of arbitrary order. The strength of derivatives of non‐integer order is their ability to describe real situations more adequately than integer order derivatives, especially when the problem has memory or hereditary properties. This paper is mainly concerned with the square‐mean almost automorphic mild solutions to a class of fractional neutral stochastic integro‐differential equations with infinite delay driven by Poisson jumps. The existence of square‐mean almost automorphic mild solutions of the previous fractional dynamical system is proved by using the method of successive approximation. The results are formulated and proved by using the fractional calculus, solution operator, and stochastic analysis techniques. Further, the existence of optimal control of the proposed problem is also presented. An example is provided to illustrate the developed theory. Copyright 2015 John Wiley & Sons, Ltd.