Maximum likelihood estimation of hidden Markov processes

Abstract

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 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 $\mathcal{F}^{Y}.$ 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.