Pointwise Second Order Necessary Conditions for Stochastic Optimal Control with Jump Diffusion

Stochastic maximum principle is one of the important major approaches to discuss stochastic control problems. A lot of work has been done on this kind of problem, see, for example, Bensoussan [3], Cadenillas and Karatzas [10], Kushner [31], Peng [41]. Recently, another kind of stochastic maximum principle, pointwise second order necessary conditions for stochastic optimal controls has been established and studied for its applications in the financial market by Zhang and Zhang [58] when the control region is assumed to be convex. In Zhang and Zhang [59], the authors extended the pointwise second order necessary conditions for stochastic optimal controls in the general cases when the control region is allowed to be non convex. Second order necessary conditions for optimal control with recursive utilities was proved by Dong and Meng [13]. In this thesis, we generalizes the work of Zhang and Zhang [58] for jump diffusions, we establish a second order necessary conditions where the controlled system is described by a stochastic differential systems driven by Poisson random measure and an independent Brownian motion. The control domain is assumed to be convex. Pointwise second order maximum principle for controlled jump diffusion in terms of the martingale with respect to the time variable is proved. The proof of the main result is based on variational approach using the stochastic calculus of jump diffusions and some estimates on the state processes. Our stochastic control problem provides also an interesting models in many applications such as economics and mathematical finance