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Martingale problems for conditional distributions of Markov processes. (English) Zbl 0907.60065

Summary: Let \(X\) be a Markov process with generator \(A\) and let \(Y(t)=\gamma (X(t))\). The conditional distribution \(\pi_t\) of \(X(t)\) given \(\sigma (Y(s):s\leq t)\) is characterized as a solution of a filtered martingale problem. As a consequence, we obtain a generator/martingale problem version of a result of L. C. G. Rogers and J. W. Pitman [Ann. Probab. 9, 573-582 (1981; Zbl 0466.60070)] on Markov functions. Applications include uniqueness of filtering equations, exchangeability of the state distribution of vector-valued processes, verification of quasireversibility, and uniqueness for martingale problems for measure-valued processes. New results on the uniqueness of forward equations, needed in the proof of uniqueness for the filtered martingale problem are also presented.

MSC:

60J25 Continuous-time Markov processes on general state spaces
60J35 Transition functions, generators and resolvents
93E11 Filtering in stochastic control theory
60G09 Exchangeability for stochastic processes
60G44 Martingales with continuous parameter
60G35 Signal detection and filtering (aspects of stochastic processes)

Citations:

Zbl 0466.60070
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