The approximation of stochastic partial differential equations and applications in nonlinear filtering.

*(English)*Zbl 0711.60053The author considers the perturbed stochastic evolution equation
\[
du_ t=Au_ tdt+F dH_ t+(B_ iu_ t+G_ i)dM^ i_ t
\]
and the perturbed Cauchy problem
\[
dv_ t=({\mathcal L}v_ t+f)dH_ t+({\mathcal M}_ iv_ t+g_ i)dM^ i_ t,\quad v_ 0\quad fixed,
\]
where A, \(B_ i\) (1\(\leq i\leq d)\) are unbounded linear operators, \({\mathcal L}\) is a second order differential operator, \({\mathcal M}_ i\) (1\(\leq i\leq d)\) are first order differential operators, H is an increasing process and \(M^ i\) (1\(\leq i\leq d)\) are continuous semimartingales.

This paper extends to the general situation (i.e. A is dissipative and \({\mathcal L}\) is degenerate) previous results of the author. A Stroock- Varadhan’s type theorem for the support of the two equations is obtained: the support is described as the closure of the paths for the related control equations. These results are applied to the Zakaî equation of the nonlinear filtering theory in order to obtain the description of the support of the unnormalized conditional density on the space \({\mathcal C}([0,T],W^ m_ 2(r))\).

This paper extends to the general situation (i.e. A is dissipative and \({\mathcal L}\) is degenerate) previous results of the author. A Stroock- Varadhan’s type theorem for the support of the two equations is obtained: the support is described as the closure of the paths for the related control equations. These results are applied to the Zakaî equation of the nonlinear filtering theory in order to obtain the description of the support of the unnormalized conditional density on the space \({\mathcal C}([0,T],W^ m_ 2(r))\).

Reviewer: M.Chaleyat-Maurel

##### MSC:

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

60G35 | Signal detection and filtering (aspects of stochastic processes) |

93E11 | Filtering in stochastic control theory |

##### Keywords:

perturbed stochastic evolution equation; perturbed Cauchy problem; Zakaî equation of the nonlinear filtering theory; unnormalized conditional density
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\textit{I. Gyöngy}, Comput. Math. Appl. 19, No. 1, 47--63 (1990; Zbl 0711.60053)

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##### References:

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