Change of Drift via Measure

Question

Let $X_t,Y_t$ be a diffusion processes under the probability measure $\mathbb{P}$ satisfying $$ dX_t = \mu(t,X_t)dt+\sigma(t,X_t)dW_t $$

$$ dY_t = \alpha(t,X_t)dt+\sigma(t,X_t)dW_t. $$ What would the Radon-Nikodym process be to make $X_t$ follow $Y_t$-dynamics?

Answer 1

Girsanov's theorem answers your question. If you can find a process $u(t,X_t)$ (in $\mathcal W_{\mathcal H}$) such that $\sigma u = \mu-\alpha$, define $$M_t:=\exp(-\int_0^t u(s,X_s)dW_s - \frac 12 \int_0^t u(s,X_s)^2ds). $$ If $M_t$ is a martingale (which holds if $u$ satisfies Novikov condition), then it is the Radon-Nikodym derivative you want.