# How to use Girsanov theorem for complicated RN derivatives?

Let $$W_t$$ be a Brownian motion under probability measure $$\mathbb{P}$$. Let $$X_t$$ be defined as follows.

$$\mathrm{d}X_t = a \mathrm{d}t + 2\sqrt{ X_t} \mathrm{d}W_t.$$

Also define: $$L_t = \exp\left(-\frac{k}{2}\int_0^t \sqrt{X_s}\mathrm{d}W_s-\frac{k^2}{8}\int_0^t X_s\mathrm{d}W_s\right).$$

If $$L_t=\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}}$$ is used to change measure from $$\mathbb{P}$$ to $$\mathbb{Q}$$, what is the dynamic of process $$X_t$$ under the new probability measure $$\mathbb{Q}$$?

In this question, the form of $$L_t$$ is different from what is used in the Girsanov theorem (i.e., the Doleans-Dade exponential). How do we use the theorem to change the measure in this case?

• $L(t)$ is not a martingale under $P$ and doesn't form an admissible radon nikodym derivative. It is exponent raised to the power a (ito integral) martingale, which isn't a martingale. Jun 12, 2021 at 4:28
• I suspect there is a typo, and that the $dW_s$ in the right should be $dt$, so that this forms the Doleans-Dade expotential. Jun 12, 2021 at 4:29