zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Maximum likelihood estimation of hidden Markov processes. (English) Zbl 1035.62084

Summary: We consider the process dY t =u t dt+dW t , where u is a process not necessarily adapted to 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 a Brownian measure. This representation involves only one basic filter (expectation of u conditional on the observed process Y). 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 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 E-M 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.

62M05Markov processes: estimation
60J60Diffusion processes
60J25Continuous-time Markov processes on general state spaces