We work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$. Let $W_t$ be a Brownian Motion adapted and $\theta(t)$ an appropriate $\mathcal{F}_t$-adapted process. According to Girsanov theorem, the following process:
$$\tilde{W}_t=W_t+\int_0^t\theta(t)\text{d}t$$
is a Brownian Motion under the measure $\tilde{\mathbb{P}}$ induced by the Doléans-Dade exponential of $\theta(t)$.
My question is, if we assume that the adapted process is a function of the Brownian Motion, $\theta(t,W_t)$, does Girsanov theorem still hold?
I don't see anything in Girsanov theorem's statement that would imply the opposite but I want to be sure.