Existence of solutions and approximate controllability of second-order stochastic differential systems with Poisson jumps and finite delay

Stochastic differential equations with Poisson jumps become very popular in modeling the phenomena arising in various fields, for instance in financial mathematics, where the jump processes are widely used to describe the asset and commodity price dynamics. The objective of this paper is to investigate the approximate controllability for a class of control systems represented by second-order stochastic differential equations with time delay and Poisson jumps. The main technique is the theory of fundamental solution constructed through Laplace transformation. By employing the so-called resolvent condition, theory of cosine operators and stochastic analysis, we formulate and prove some sufficient conditions for the approximate controllability of the considered system. In the end an example is given and discussed to illustrate the obtained results.