Second‐Order Maximum Principle for Optimal Controls for Stochastic Systems With Teugels Martingales Under Partial Information

ABSTRACT

In this article, under partial information, we establish a second‐order stochastic maximum principle for optimal stochastic control. The system is governed by nonlinear controlled Itô‐type stochastic differential equations driven by orthogonal Teugels martingales associated with a Lévy process, and an independent Brownian motion. We establish pointwise second‐order necessary conditions for the optimal control under incomplete information. The control domain is assumed to be bounded and convex. It is required that the control process is adapted to a given subfiltration of the filtration generated by the underlying Lévy processes. The proof of our main result is based on a variational approach using the stochastic calculus of orthogonal Teugels martingales. Finally, an example is studied to illustrate our theoretical results.