Spatial-Temporal Nonlinear Filtering for Poisson Random Fields

The article considers the problem of estimating a diffusion process $\mathbf x_t$ from results of observations of a Poisson random field $N(t,\xi)$ whose parameter (intensity function) depends on the process $\mathbf x_t$. On the basis of Grigelionis’ results [Lit. Mat. Sb., 1972, vol. 12, no. 4, pp. 37–51] the authors obtain a stochastic equation for the probability density of $\mathbf x_1$; expressions are derived in the Gaussian approximation for the estimate ${\mathbf x}^{\widehat{}}_t$ and covariation matrix $\mathbf D_t$; two examples are considered.