Second-order necessary condition for partially observed stochastic system with random jumps

In this paper, we investigate the singular optimal control problem for partially observed stochastic control system with random jumps. In our model, the control variable is allowed to enter all coefficients of the state equation and observation equation and the control domain is assumed to be convex. In this case, we obtain the integral -type second -order necessary condition for singular optimal control in the classical sense. Furthermore, under some regularity assumption, we establish the pointwise second -order necessary condition for singular optimal control by virtue of Malliavin calculus. Finally, we present two examples to demonstrate the important significance of our results.