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Maximum likelihood estimation of hidden Markov processes. (English) Zbl 1035.62084

Summary: We consider the process dY t =u t dt+dW t , where u is a process not necessarily adapted to Y (the filtration generated by the process Y) and W is a Brownian motion. We obtain a general representation for the likelihood ratio of the law of the Y process relative to a Brownian measure. This representation involves only one basic filter (expectation of u conditional on the observed process Y). This generalizes a result of T. Kailath and M. Zakai [Ann. Math. Stat. 42, 130–140 (1971; Zbl 0226.60061)] where it is assumed that the process u is adapted to Y .

In particular, we consider the model in which u is a functional of Y and of a random element X which is independent of the Brownian motion W. For example, X could be a diffusion or a Markov chain. This result can be applied to the estimation of an unknown multidimensional parameter θ appearing in the dynamics of the process u based on continuous observation of Y on the time interval [0,T].

For a specific hidden diffusion financial model in which u is an unobserved mean-reverting diffusion, we give an explicit form for the likelihood function of θ. For this model we also develop a computationally explicit E-M algorithm for the estimation of θ. In contrast to the likelihood ratio, the algorithm involves evaluation of a number of filtered integrals in addition to the basic filter.

MSC:
62M05Markov processes: estimation
60J60Diffusion processes
60J25Continuous-time Markov processes on general state spaces