When Does the Ramer Formula Look Like the Girsanov Formula?

Abstract

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.