Linear-quadratic optimal control for partially observed forward-backward stochastic systems with random jumps

In this paper, we investigate a linear-quadratic (LQ) optimal control problem for partially observed forward-backward stochastic systems with random jumps, where the observation’s drift term is linear with respect to the state x and control variable v. In our model, the observation process is no longer a Brownian motion but a controlled stochastic process driven by Brownian motions and Poisson random measures, which also have correlated noises with the state equation. Applying a backward separation approach to decompose the state and observation, we overcome the problem of cyclic dependence of control and observation. Then, the necessary and sufficient conditions for optimal control are derived. We also obtain the feedback representation of optimal control and provide two special cases to illustrate the significance of our results. Moreover, we also provide a financial application to demonstrate the practical significance of our results.