Roughly speaking, Girsanov's theorem says that if we have a Brownian motion $W$ on $[0,T]$, we can construct a new process with a modified drift that has an equivalent law to $W$ (subject to adaptedness constraints on the drift, integrability, etc...).

I'm wondering if there's a 'converse' to Girsanov's theorem. If we have some process $X$ whose law $\mathbb{P}$ is equivalent to Wiener measure $\mathbb{W}$, can we always express $X$ as $X_t = \alpha_t + W_t$, where $\alpha$ is adapted to the filtration of $W$? Alternatively, can the Radon-Nikodym derivative always be expressed in the form $$\frac{d\mathbb{P}}{d\mathbb{W}} = \exp\left( \int \alpha_u dW_u - \frac{1}{2}\int \alpha^2_u du \right)$$.