On the Relation of Zakai’s and Mortensen’s Equations

Abstract

The problem of optimal control for partially observable diffusion processes is studied by the dynamic programming in function space approach first proposed by R. E. Mortensen. The density of the conditional distribution of the (unobservable) signal given past and present observations, which satisfies the Zakai equation of nonlinear filtering, is viewed as the new state of the system. A verification result is established for the corresponding Bellman–Hamilton–Jacobi equation, known as Mortensen’s equation.