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

Summary: We consider the process $d{Y}_{t}={u}_{t}dt+d{W}_{t}$, where $u$ is a process not necessarily adapted to ${ℱ}^{Y}$ (the filtration generated by the process $Y\right)$ 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 $Y\right)$. 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 ${ℱ}^{Y}$.

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 $\left[0,T\right]$.

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 60J60 Diffusion processes 60J25 Continuous-time Markov processes on general state spaces