Evaluation of likelihood-ratio and performance bounds for nonlinear decision problems via stochastic PDE

The nonlinear binary decision problem with signal satisfying a diffusion equation observed through noisy measurements is considered. Using the unnormalized conditional density of nonlinear filtering, expressions for evaluating the decision strategy and error probability bounds are obtained, in terms of the solution of a stochastic partial differential equation. This equation is solvable for linear as well as certain nonlinear decision problems. In the limit as the random inputs tend to zero, the application of large deviations result for the unnormalized conditional distribution gives asymptotic estimates for evaluating the decision strategy and performance.