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An existenc theorem for a stochastic partial differential equation arising from filtering theory. (English) Zbl 0557.60045

The Cauchy problem for the following stochastic partial differential equation \[ du(t,x)=u_{xx}(t,x)dt+h(x)u(t,x)dw_ t,\quad u(0,x)=u_ 0(x) \] where h is any polynomial of degree n and w is a real Wiener process, is investigated. The method consists in performing a transformation of the problem by \[ v(t,x)=\exp [-h(x)w(t)]u(t,x) \] to get a deterministic equation w.p. 1 with respect to v(t,x).
Reviewer: S.Eloshvili

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
93E11 Filtering in stochastic control theory

Keywords:

Cauchy problem
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References:

[1] G. Da Prato - M. Iannelli - L. Tubaro , Some results on linear stochastic differential equations in Hilbert spaces , Stochastic , 6 ( 1982 ), pp. 105 - 116 . MR 665246 | Zbl 0475.60041 · Zbl 0475.60041 · doi:10.1080/17442508208833210
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