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http://hdl.handle.net/2451/14750
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| Title: | MAXIMUM LIKELIHOOD ESTIMATION OF HIDDEN MARKOV PROCESSES |
| Authors: | Frydman, Halina
Lakner, Peter |
| Issue Date: | 2001 |
| Publisher: | Stern School of Business, New York University |
| Series/Report no.: | SOR-2001-3 |
| Abstract: | We consider the process dYt = ut dt + dWt , where u is a process not
necessarily adapted to 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) 130â140] 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 θ
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
θ. For this model we also develop a computationally
explicit EâM algorithm for the estimation of θ. In
contrast to the likelihood ratio, the algorithm involves evaluation of a
number of filtered integrals in addition to the basic filter. |
| URI: | http://hdl.handle.net/2451/14750 |
| Appears in Collections: | IOMS: Statistics Working Papers
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