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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 $${\mathcal F}^Y$$ (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 $${\mathcal F}^Y$$.
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 $$\theta$$ 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 $$\theta$$. For this model we also develop a computationally explicit E-M algorithm for the estimation of $$\theta$$. In contrast to the likelihood ratio, the algorithm involves evaluation of a number of filtered integrals in addition to the basic filter.

##### MSC:
 62M05 Markov processes: estimation; hidden Markov models 60J60 Diffusion processes 60J25 Continuous-time Markov processes on general state spaces
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