  This paper is concerned with a discounted optimal control problem of
partially observed forward-backward stochastic systems with jumps on infinite
horizon. The control domain is convex and a kind of infinite horizon
observation equation is introduced. The uniquely solvability of infinite
horizon forward (backward) stochastic differential equation with jumps is
obtained and more extended analysis, especially for the backward case, is made.
Some new estimates are first given and proved for the critical variational
inequality. Then an ergodic maximum principle is obtained by introducing some
infinite horizon adjoint equations whose uniquely solvabilities are guaranteed
necessarily. Finally, some comparison are made with two kinds of representative
infinite horizon stochastic systems and their related optimal controls.
