Open Access
July, 1992 When Does the Ramer Formula Look Like the Girsanov Formula?
M. Zakai, O. Zeitouni
Ann. Probab. 20(3): 1436-1440 (July, 1992). DOI: 10.1214/aop/1176989698


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.


Download Citation

M. Zakai. O. Zeitouni. "When Does the Ramer Formula Look Like the Girsanov Formula?." Ann. Probab. 20 (3) 1436 - 1440, July, 1992.


Published: July, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0762.60029
MathSciNet: MR1175269
Digital Object Identifier: 10.1214/aop/1176989698

Primary: 60G30
Secondary: 47B10 , 60H07

Keywords: Absolute continuity , Girsanov formula , quasinilpotent operators , Ramer formula

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 3 • July, 1992
Back to Top