Existence, Uniqueness and Asymptotic Behavior of Solutions to a Class of Zakai Equations with Unbounded Coefficients

Conditions are given to guarantee the existence and uniqueness of solutions to the Zakai equation associated with the nonlinear filtering of diffusion processes. The conditions permit stronger than a polynomial growth of the coefficients and depend instead on the relative growth rates. The results are derived by adapting, through a sequence of exponential transformations, the classical existence and uniqueness theorems for parabolic PDE's due to Besala to the "robust" form of the Zakai equation. In this process, we also obtain sharp estimates for the tail behavior of the conditional density. Examples, including observations through a polynomial sensor and estimation of the state of a "bilinear" system, are worked out in detail. Our results are compared to those of Fleming and Mitter, Pardoux, and Sussmann who, among others, have obtained existence and uniqueness theorems for a more limited class of problems by different methods.