Robust smoother dynamics for Poisson processes driven by an Ito^diffusion

Abstract

We compute general smoothing dynamics for partially observed dynamical systems with Poisson observations. The model we consider is a Markov modulated Poisson processes, whose intensity depends upon the state of unobserved Itoˆ process. All smoother dynamics depend, in some manner, upon filtered estimates of an unobserved state process. To compute filtered estimates of state, we establish a Duncan-Mortenson-Zakai (DMZ) equation, however, this filter includes a stochastic integration. By adapting the transformation techniques developed in Clark (1978) and Davis (1980) we compute robust a form of the DMZ equation which does not include stochastic integration. To construct smoothers, we exploit a duality between forward and backwards dynamics. Smoothed state estimates are computed by using the forward and backwards robust equations. The general smoother dynamics we present can readily be applied to specific smoothing algorithms, referred to in the literature as: fixed point smoothing, fixed lag smoothing and fixed interval smoothing.