Robust detection filters for jump Markov systems with doubly stochastic Poisson process models

In this article we consider a dynamic M-ary detection problem when Markov chains are observed through a doubly stochastic Poisson process. These systems are fully specified by a candidate set of parameters, whose elements are, a rate matrix for the Markov chain and a vector of Poisson intensities for the observation model. Further, we suppose these parameter sets can switch according to the state of an unobserved Markov chain and thereby produce an observation process generated by time varying (jump stochastic) parameter sets. Given such an observation process and an assumed collection of models, we compute a filter whose solution is the estimated probabilities of each model parameter set explaining the observation. By defining a new augmented state process, then applying the method of reference probability, we compute matrix-valued dynamics whose solutions estimate joint probabilities for all combinations of candidate model parameter sets, and values taken by the indirectly observed state process. These matrix-valued dynamics satisfy a stochastic integral equation with a Lebesgue-Stieltjes integrator. Using the gauge transformation techniques, we compute robust matrix-valued dynamics for the joint probabilities on the augmented state space. In these new dynamics the observed Poisson process appears as a parameter in the fundamental matrix of a linear ordinary differential equation, rather than an integrator in a stochastic integral equation.