The Annals of Probability

When Does the Ramer Formula Look Like the Girsanov Formula?

M. Zakai and O. Zeitouni

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Let $\{B,H,P_0\}$ be an abstract Wiener space and for every real $\rho$, let $T_\rho\omega = \omega + \rho F(\omega)$ be a transformation from $B$ to $B$. It is well known that under certain assumptions the measures induced by $T_\rho$ or $T_\rho^{-1}$ are mutually absolutely continuous with respect to $P_0$ and the density function is represented by the Ramer formula. In this formula, the Carleman-Fredholm determinant $\det_2(I_H + \rho\nabla F)$ appears as a factor. We characterize the class of $\nabla F$ for which a.s.-$P_0, \det_2(I_H + \rho\nabla F) = 1$ for all $\rho$ in an open subset of $\mathbb{R}$, in which case the form of Ramer's expression reduces to the familiar Cameron-Martin-Maruyama-Girsanov form. The proof is based on a characterization of quasinilpotent Hilbert-Schmidt operators.

Article information

Ann. Probab., Volume 20, Number 3 (1992), 1436-1440.

First available in Project Euclid: 19 April 2007

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Primary: 60G30: Continuity and singularity of induced measures
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 47B10: Operators belonging to operator ideals (nuclear, p-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20]

Girsanov formula Ramer formula absolute continuity quasinilpotent operators


Zakai, M.; Zeitouni, O. When Does the Ramer Formula Look Like the Girsanov Formula?. Ann. Probab. 20 (1992), no. 3, 1436--1440. doi:10.1214/aop/1176989698.

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