I wish to prove the following the statement:
"If $\mu$ and $\gamma$ are probability measures on $C([0,\infty), \mathbb{R}^d)$, with $\gamma$ being the standard Weiner measure, $W_t$ being standard brownian motion, and $d \mu = F d \gamma$, then there exists an adapted process $u_t$ satisfying $\int_0^\infty u^2_s ds < \infty$ $\mu$-almost surely, such that $W^u_t := W_t - \int_0^t u_s ds$ is Brownian motion under $\mu$. "
If we pay no attention to detail, the proof should go as follows:
Let $F_t = \mathbb{E}[F | \mathcal{G}_t]$, where $\mathcal{G}_t$ be the Brownian filtration. Then it is a $\gamma$-martingale, and so by martingale representation theorem, we have $dF_t = \langle M_t, dW_t \rangle$. Standard martingale argument gives you that $\mu ( \inf_{t > 0} F_t > 0) = 1$ so it's okay to define $u_t = M_t / F_t$. We have then that $F_t = \exp( \int_0^t \langle u_s, dW_s \rangle - \frac{1}{2} \int_0^t |u_s|^2 ds)$ and so the applying the Girsanov theorem yields us what we want.
My questions are:
1) Why should our chosen $u_s$ satisfy $\int_0^{\infty} |u_s|^2 ds < \infty$ $\mu$-almost surely? I can get that $ \int_0^{\infty} |F_s u_s|^2 ds < \infty$ $\gamma$-almost surely. I think I should be able to swap the $\gamma$ for a $\mu$ at the cost of $F_s$, but I have $F_s$^2, which sorta ruins my day.
2) Related to 1), I know that if we DID have $\int_0^{\infty} |u_s|^2 ds < \infty$, then by classical theory, $F_t$ is a martingale. However, we already that $F_t$ is a martingale. I doubt the statement goes the other way (but if it does, this would nicely solve 1) !), but how much information can we obtain about $u_s$ just knowing that $F_t$ is a martingale?
3) Most references that deal with these kind of issues tend to assume that the paths are from $[0,T]$, where $T$ is finite, presumably so that they can use some uniform integrability arguments, and so that the Novikov condition works if you assume $u_s$ is bounded. How much (if anything) is lost if you take paths from $[0,\infty)$ and say that e.g. Novikov's condition holds for each fixed $T > 0$?
Thank you for all your help!