A Stochastic Maximum Principle for Partially Observed Jump-Diffusion Systems with State-Dependent Counting-Process Observations

Abstract

This paper studies a partially observed stochastic control problem for jump-diffusion state processes observed through multivariate counting processes with state-dependent intensities. In contrast to diffusion observations or observation jumps with state-independent intensities, each observation jump carries information about the latent state and enters the likelihood-ratio dynamics, producing a coupled variational structure involving both the state perturbation and the likelihood-ratio perturbation. By introducing a reference probability measure and augmenting the state with the counting-process likelihood ratio, we derive a stochastic maximum principle for the resulting partially observed control problem. The necessary optimality condition is expressed as a conditional Hamiltonian stationarity relation with respect to the observation filtration. As an illustration, we apply the result to a linear-quadratic execution model and obtain a Riccati-type feedback driven by the conditional mean of the latent state. The resulting feedback is illustrated numerically by a particle-filter implementation under nonlinear point-process filtering.