If we create a weak solution of an SDE using the Girsanov transformation, are the initial condition and parameters independent of the transformed Wiener process if they are independent of the original Wiener process? This other question is similar but more general.
Let $(\Omega, \mathcal{F}, P)$ be a probability space and $X_t$ be a stochastic process which satisfies the following SDE:
$$ d X_t = f(t, X_t,\Theta)dt + dW_t, $$
where $W_t$ is a Wiener process and $\Theta$ is a random variable. We assume $W_t$ be adapted to the filtration $\mathcal{F}_t$ and that $\Theta$ and the initial condition of the SDE $X_0$ are $\mathcal{F}_0$-measurable and independent of the $\sigma$-algebra generated by $W$.
If we define a stochastic process $M_t$ by $$ M_t := \exp\left[-\int_0^t f(s, X_s,\Theta) \,dW_s -\frac{1}{2}\int_0^t f(s, X_s,\Theta)^2 \,ds\right] $$ and a new measure $\tilde P(A):=\int_A M_T dP$, then the process $X_t$ also satisfies $d X_t = d\tilde W_t$ where the process $\tilde W_t := \int_0^t f(s,X_s,\Theta)\,ds + W_t$ is a Wiener process under $\tilde P$. Under the measure $\tilde P$, are $X_0$ and $\Theta$ independent of $\tilde W$?
I know that the $\tilde P(A)=P(A)$ for all $A\in\mathcal{F}_0$ because $M_t$ is an $\mathcal{F}_t$ martingale and $M_0=1$, so the measures induced by $X_0$ and $\Theta$ do not change. I believe I would be able to prove the independence if it is possible to show that $$ P(W^{-1}(A)|\mathcal{F}_t) = \tilde P(\tilde W^{-1}(A)|\mathcal{F}_t) \qquad\text{with prob. 1}, $$ for any set $A$ from the Borel $\sigma$-algebra of the classical Wiener space. Any ideas?
The answer is that the proposition is true, that is, any $\mathcal{F}_0$-measurable random variable is independent of $\tilde W$ under $\tilde P$, and the answer is quite simple. It was simply a matter of finding a more complete statement of the Girsanov theorem / transformation.
By definition, a Wiener process on a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{0\leq t\leq T}, P)$ has $W_t-W_s$ independent of $\mathcal{F}_s$ for all $0\leq s \leq t \leq T$. As $W_0=0$ is almost surely, this means that the process is independent of $\mathcal{F}_0$. More complete statements of the Girsanov transformation, like that of Ikeda and Watanabe, say that $\tilde W$ is a Wiener process on the filtered space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{0\leq t\leq T}, \tilde P)$, hence independent of $\mathcal{F}_0$.