Recursive nonlinear estimation of a diffusion acting as the rate of an observed Poisson process

A conditional Poisson process is observed, whose rate is a diffusion process of known structure. The problem is to estimate the rate from the observed point process. Recursive equations are given for the conditional moment generating function and for an unnormalized conditional probability density of the rate. By studying these equations separately in between jumps and at the jumps, series expansions are obtained for these generating functions and densities in a number of examples that arise in applications to optical modulation and communications networks.