  IIn this paper, we study a partially observed progressive optimal control
problem of forward-backward stochastic differential equations with random
jumps, where the control domain is not necessarily convex, and the control
variable enter into all the coefficients. In our model, the observation
equation is not only driven by a Brownian motion but also a Poisson random
measure, which also have correlated noises with the state equation. For
preparation, we first derive the existence and uniqueness of the solutions to
the fully coupled forward-backward stochastic system with random jumps and its
estimation in $L^\beta(\beta\geq2)$-space under some assumptions, and the
non-linear filtering equation of partially observed stochastic system with
random jumps. Then we derive the partially observed global maximum principle
with random jumps. To show its applications, a partially observed linear
quadratic progressive optimal control problem with random jumps is
investigated, by the maximum principle and stochastic filtering. State estimate
feedback representation of the optimal control is given in a more explicit form
by introducing some ordinary differential equations.
