Annals of Probability

Nonlinear Transformations on the Wiener Space

Ognian Enchev

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We study shift transformations on a general abstract Wiener space $(E, H, \mu)$, which have the form: $E \ni \omega \mapsto \mathscr{J}^\phi\omega \equiv \omega - \int^T_0 \phi_t(\omega)Z(dt) \in E,$ where $\phi_t(\omega)$ is a scalar function on $\lbrack 0, T\rbrack \times E$ and $Z$ is an orthogonal $H$-valued measure. Under suitable conditions for the kernel $\phi$, we construct explicitly a probability measure $\mu^\phi$ on $E$, which is equivalent to the standard Wiener measure $\mu$ and has the property: $\mu^\phi\{\mathscr{F}^\phi \in A\} = \mu(A), A \in \mathscr{B}_E$. The main result presents an analog of the well-known Cameron-Martin-Girsanov theorem for the case where the shift is allowed to anticipate. This leads to an additional integral term in the Girsanov exponent. Also, the Wiener-Ito integral in this exponent is now replaced by an extended stochastic integral.

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Ann. Probab., Volume 21, Number 4 (1993), 2169-2188.

First available in Project Euclid: 19 April 2007

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Primary: 60B05: Probability measures on topological spaces
Secondary: 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus 46G12: Measures and integration on abstract linear spaces [See also 28C20, 46T12] 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20] 47A68: Factorization theory (including Wiener-Hopf and spectral factorizations)

Abstract Wiener spaces stochastic integrals with anticipating integrands Gohberg-Krein factorization absolutely continuous transformations of the Wiener measure


Enchev, Ognian. Nonlinear Transformations on the Wiener Space. Ann. Probab. 21 (1993), no. 4, 2169--2188. doi:10.1214/aop/1176989015.

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