On second-order control problem of stochastic systems with random Lévy process

The main purpose of this paper is to derive second-order necessary conditions for optimal control, where the controlled stochastic differential system is driven by random Lévy process and an independent Brownian motion. We prove variational principle for second-order control problem in the form of Pontryagin-type maximum principle under partial information. The proof of our main result is based on some techniques of variational analysis via the stochastic calculus of random Lévy process. An example is discussed to illustrate our theoretical result.