Abstract
Problem 3 in the introduction is a special case of the following general filtering problem:
Suppose the state X t ∈ R n at time t of a system is given by a stochastic differential equation
where b: R n+1 → R n, σ : R n+1 → R n × p satisfy conditions (5.2.1), (5.2.2) and W t is p-dimensional white noise. As discussed earlier the Ito interpretation of this equation is (system)
where U t is p-dimensional Brownian motion. We also assume that the distribution of X 0 is known and independent of U t . Similarly to the 1-dimensional situation (3.3.6) there is an explicit several-dimensional formula which expresses the Stratonovich interpretation of (6.1.1):
in terms of Ito integrals as follows:
where
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© 1998 Springer-Verlag Berlin Heidelberg
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Øksendal, B. (1998). The Filtering Problem. In: Stochastic Differential Equations. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03620-4_6
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DOI: https://doi.org/10.1007/978-3-662-03620-4_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63720-2
Online ISBN: 978-3-662-03620-4
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