A Girsanov Measure-Change Question for a Square-Root Diffusion

Summary

The document poses a stochastic-calculus question about changing probability measure for a process satisfying a square-root diffusion. It specifies a proposed likelihood process and asks how the drift of the diffusion would change under the new measure, given that the likelihood is not written in the usual Doléans–Dade exponential form used in Girsanov’s theorem.

No answer or derivation is included, so the document does not establish the new dynamics or verify that the proposed likelihood is a valid density process. In particular, a reader would need to check the stochastic-integral terms and the conditions that make the likelihood a true martingale before applying Girsanov. Its value is as a focused question about recognizing and converting a change-of-measure density; it does not provide a resolved method or evidence.

Key ideas

• The question concerns a square-root diffusion driven by Brownian motion.
• It asks how to derive the process drift after a probability-measure change.
• The proposed likelihood is presented in a form that differs from the standard exponential used in Girsanov’s theorem.
• The document supplies no solution and does not establish that the proposed likelihood is a valid density process.

Tags

# How to use Girsanov theorem for complicated RN derivatives?


# 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?

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