Maximum Likelihood Estimation of Hidden Markov Processes

We consider the process dYt = ut dt + dWt , where u is a processnot necessarily adapted to F Y (the filtration generated by the process Y)and W is a Brownian motion. We obtain a general representation for thelikelihood ratio of the law of the Y process relative to Brownian measure.This representation involves only one basic filter (expectation of u conditionalon observed process Y). This generalizes the result of Kailath and Zakai[Ann.Math. Statist. 42 (1971) 130â€"140] where it is assumed that the process uis adapted to F Y . In particular, we consider the model in which u is afunctional of Y and of a random element X which is independent of theBrownian motion W. For example, X could be a diffusion or a Markov chain.This result can be applied to the estimation of an unknown multidimensionalparameter I¸ appearing in the dynamics of the process u based on continuousobservation of Y on the time interval [0,T ]. For a specific hidden diffusionfinancial model in which u is an unobserved mean-reverting diffusion, wegive an explicit form for the likelihood function of I¸. For this model we alsodevelop a computationally explicit Eâ€"M algorithm for the estimation of I¸. Incontrast to the likelihood ratio, the algorithm involves evaluation of a numberof filtered integrals in addition to the basic filter.