Hamilton-Jacobi-Bellman equations for the optimal control of the Duncan-Mortensen-Zakai equation

We study a class of Hamilton-Jacobi-Bellman #HJB# equations associated to stochastic optimal control of the Duncan-Mortensen-Zakai equation. The equations are investigated in weighted L 2 spaces. Weintroduce an appropriate notion of weak #viscosity# solution of such equations and prove that the value function is the unique solution of the HJB equation. We apply the results to stochastic optimal control problems with partial observation and correlated noise. Key words: Hamilton-Jacobi-Bellman equations, Duncan-Mortensen-Zakai equation, optimal control of partially observed systems, viscosity solutions. 1991 Mathematics Subject Classi#cation: 35R15, 49L25, 93E11, 93E20. # The authors were supported by NSF grant DMS 97-06760. 1 1 Introduction The paper is devoted to the study of Hamilton-Jacobi-Bellman #HJB# equations associated with stochastic optimal control problems for the Duncan-Mortensen-Zakai #DMZ# equation. Such HJB equations have the form 8 # # # # # # # : v t + inf #2A ..