A Stochastic Maximum Principle for a Minimization Problem Under Partial Information

In this paper, we establish a stochastic maximum principle for a stochastic minimization problem under partial information. With the Backward stochastic differential equations (in short BSDE’s), we establish a sufficient condition of optimality to characterize and determine an optimal control. This is done instead of using the Hamiltonian which is a deterministic function. The equations translating the dynamics of the state variables of the controlled system contain an BSPDE (Backward stochastic partial differential equation) which can be the unnormalized conditional density like the Zakai equation born from a problem of passage from partial to full information.