Nonlinear Transformations on the Wiener Space

Abstract

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.