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November 2003 Maximum likelihood estimation of hidden Markov processes
Halina Frydman, Peter Lakner
Ann. Appl. Probab. 13(4): 1296-1312 (November 2003). DOI: 10.1214/aoap/1069786500


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.


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Halina Frydman. Peter Lakner. "Maximum likelihood estimation of hidden Markov processes." Ann. Appl. Probab. 13 (4) 1296 - 1312, November 2003.


Published: November 2003
First available in Project Euclid: 25 November 2003

zbMATH: 1035.62084
MathSciNet: MR2023878
Digital Object Identifier: 10.1214/aoap/1069786500

Primary: 60J25 , 60J60 , 62M05

Keywords: E-M algorithm , filtered integrals , Hidden diffusion financial models , likelihood ratio , maximum likelihood estimation

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.13 • No. 4 • November 2003
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