# Change of Drift via Measure

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?

• Doesn't Girsanov theorem give directly the answer to this kind of problem what have you tried ? Commented Dec 13, 2017 at 10:40
• Dear @TheBridge could you maybe give a reference on why Girsanov's theorem is exactly what we need here? I have read the statement several times, but I do not see the connection.. And in this question, for the process $Y_t$ he probably meant some other $\tilde{\sigma}(t,Y_t)$ for the second term, right? And assuming this, the answer below still holds, doesn't it? Thank you. Commented Apr 13, 2023 at 7:05

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.