The Maximum Principle for Partially Observed Optimal Control of FBSDE Driven by Teugels Martingales and Independent Brownian Motion

The aim of this paper is to study a stochastic partially observed optimal control problem, for systems of forward backward stochastic differential equations (FBSDE for short), which are driven by both a family of Teugels martingales and an independent Brownian motion. By using Girsavov’s theorem and a standard spike variational technique, we prove necessary conditions to characterize an optimal control under a partial observation, where the control domain is supposed to be convex. Moreover, under some additional convexity conditions, we prove that these partially observed necessary conditions are sufficient. In fact, compared to the existing methods, we get the last achievement in two different cases according to the linearity or the nonlinearity of the terminal condition for the backward component. As an illustration of the general theory, an application to linear quadratic control problems is also investigated.