I’m looking for a version of the Cameron Martin theorem for the Brownian motion under random shifts. Here is the precise statement:
Let $\mathbb P$ be Wiener measure on $\Omega := C[0, 1]$. Given a $C[0, 1] $ valued random variable $F$, define the translation map $T_F: \Omega \to \Omega$ by $T_F (\omega) = \omega + F(\omega)$, and denote the pushforward of $\mathbb P$ under this map by $T_F^{\ast} \, \mathbb P $.
Denote by $W^{1, 2}$ the space of absolutely continuous functions on $[0, 1]$ with derivative in $L^2$.
Theorem: $\mathbb Q$ is equivalent to $\mathbb P$ if and only if $\mathbb Q = T_F ^\ast \, \mathbb P$ for some $F$ such that $F(0) = 0$ and $F \in W^{1, 2}$ almost surely.
My questions are two-fold:
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Is this statement true?
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If so, where can I find a proof?
So far the statements I could find were only for deterministic
shifts $F$, these are what is usually called the Cameron Martin theorem
for Brownian motion. On the other hand, statements of Girsanov’s theorem deal with random shifts, but typically are concerned with the Radon Nikodym derivative, and are silent on the interpretation as shifts by $W^{1, 2}$ processes.
Best Answer
Let $\varphi$ be a smooth function such that $\varphi(x)=0$ for $x\leq 0$ and $\varphi(x)=1$ for $x\geq 1$, and $0\leq\varphi(x)\leq 1$ else. If $\tau_1$ and $\tau_2$ are is the first times $B_t$ hits $1,2$ respectively, and $M$ is a maximum of the $B_t$ on $[0,1]$, put $F_t=-M\varphi(\frac{t-\tau_1}{\tau_2-\tau_1})$. Then, $\sup (B_t+F_t)\leq 2$ almost surely, hence $\mathbb{P}$ is not absolutely continuous with respect to $\mathbb{Q}$.