Given an underlying filtered probability space $\left( \Omega,\left\{ \mathcal{F}_t \right\}_{t\in[0,T]},P \right)$ that satisfies the usual conditions, define an OU process with
$$ dY(t)=-\kappa Y(t)\,dt+\sigma \,dZ_t $$ where $\kappa$ and $\sigma>0$ are some parameters. $dZ_t$ is a $1$-dimensional standard Brownian motion. Loosely put, I would like to think of $Y(t)$ as the degree to which changing measures shifts the Brownian motion, to be specified below.
Then the solution is $Y(t) = \int_0^t e^{-\kappa(t-s)}\sigma \, dZ_s$ for a zero initial condition.
Also define geometric Brownian motion
$$ \frac{dX(t)}{X(t)}=\mu \,dt+\sigma \,dZ_t $$
that is our main process of interest.
Further define a process $\xi_t$ where
$$ \frac{d\xi_t}{\xi_t}=Y(t) \, dZ_t $$ that I would like to think of as the density process to be specified later. The solution is
$$ \xi_t =\exp\left(\int_0^t Y(s)\,dw(s)-\frac{1}{2} \int_0^t Y(s)^2 \, ds\right) $$
I would like to know if I can construct a measure $Q$ (equivalent to $P$) and the corresponding Radon Nikodym derivative $dQ/dP$ where:
$X(t)$ under measure $Q$ evolves as $$ \frac{dX(t)}{X(t)}=(\mu+\sigma Y(t)) \, dt+\sigma \, dZ_t^Q $$ where $dZ_t^Q = dZ_t -Y(t)\,dt$ is a Brownian motion under $Q$.
$\xi_0 =1,$ for $0\leq t\leq T$ $$ \xi_t=\mathbb{E}\left[\left.\frac{dQ}{dP}\right|\mathcal{F}_t\right] $$ so that $\xi_T = \frac{dQ}{dP}$.
The main issue for me is that $Y(t)$ is not some deterministic function, but a random process that depends on the path of Brownian motion.
