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 L2 spaces. We introduce 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.