Maximum principle for backward stochastic systems associated with Lévy processes under partial information

This paper studies an optimal control problem for a backward stochastic control systems associated with Lévy processes under partial information. More precisely, the controlled systems are described by backward stochastic differential equations driven by Teugels martingales and an independent multi-dimensional Brownian motion, where Teugels martingales are a family of pairwise strongly orthonormal martingales associated with Lévy processes, and all admissible control processes are required to be adapted to a given subfiltration of the filtration generated by the underlying Teugels martingales and Brownian motion. For this type of partial information stochastic optimal control problem with convex control domain, we derive the necessary and sufficient conditions for the existence of the optimal control by means of convex analysis and duality techniques. As an application, the optimal control problem of linear backward stochastic differential equation with a quadratic cost criteria (called backward linear-quadratic problem, or BLQ problem for short) under partial information is discussed and the unique optimal control is characterized explicitly by adjoint processes.