Maximum Likelihood Estimation of Hidden Markov Processes
نویسندگان
چکیده
We consider the process dYt = utdt + dWt; where u is a process not necessarily adapted to FY (the ...ltration 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 ...lter (expectation of u conditional on observed process Y ): This generalizes the result of Kailath and Zakai (1971) 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 di¤usion or a Markov chain. This result can be applied to the estimation of an unknown multidimensional parameter μ appearing in the dynamics of the process u based on continuous observation of Y on the time interval [0,T]. For a speci...c hidden di¤usion ...nancial model in which u is an unobserved mean-reverting di¤usion, we give an explicit form for the likelihood function of μ: For this model we also develop a computationally explicit E-M algorithm for the estimation of μ: In contrast to the likelihood ratio, the algorithm involves evaluation of a number of ...ltered integrals in addition to the basic ...lter.
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