Maximum likelihood estimation of hidden Markov processes

We consider the process dYt=utdt+dWt, where u is a process not necessarily adapted to FY (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 Brownian measure. This representation involves only one basic filter (expectation of u conditional on observed process Y). This generalizes the result of Kailath and Zakai [Ann. Math. Statist. 42 (1971) 130-140] where it is assumed that the process u is adapted to FY. 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 . 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 .. For this model we also develop a computationally explicit E--M algorithm for the estimation of .. In contrast to the likelihood ratio, the algorithm involves evaluation of a number of filtered integrals in addition to the basic filter.