Robust M-ary detection filters for continuous-time jump Markov systems

In this article we consider a dynamic M-ary detection problem when Markov chains are observed through a Wiener process. These systems are fully specified by a candidate set of parameters, whose elements are: a rate matrix for the Markov chain and a parameter 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. We estimate the probabilities of each model parameter set explaining the observation. Using the gauge transformation techniques introduced by Clark (1977) and a pointwise matrix product, we compute robust matrix-valued dynamics for the joint probabilities on the augmented state space. In these new dynamics the observation Wiener process appears as a parameter in the fundamental matrix of a linear ordinary differential equation, rather than an integrator in a stochastic integral equation. Finally, by exploiting a duality between causal and anticausal robust detector dynamics, we develop an algorithm to compute smoothed mode probability estimates without stochastic integrations.