Existence of solutions and approximate controllability of impulsive fractional stochastic differential systems with infinite delay and Poisson jumps

The paper is motivated by the study of interesting models from economics and the natural sciences where the underlying randomness contains jumps. Stochastic differential equations with Poisson jumps have become very popular in modeling the phenomena arising in the field of financial mathematics, where the jump processes are widely used to describe the asset and commodity price dynamics. This paper addresses the issue of approximate controllability of impulsive fractional stochastic differential systems with infinite delay and Poisson jumps in Hilbert spaces under the assumption that the corresponding linear system is approximately controllable. The existence of mild solutions of the fractional dynamical system is proved by using the Banach contraction principle and Krasnoselskii’s fixed-point theorem. More precisely, sufficient conditions for the controllability results are established by using fractional calculations, sectorial operator theory and stochastic analysis techniques. Finally, examples are provided to illustrate the applications of the main results.