# Stochastic integrals and new probability measures

Let $B$ be a standard Brownian motion on $(\Omega, \mathcal{F}, P, ({\mathcal{F}_t})_{t\ge0})$, where the filtration is the one generated by $B$. Fix a time interval $[0,T]$. Define the process $X$ as the solution to the SDE $$\mathrm dX_t = \sigma X_t\,\mathrm dB_t,\quad X_0 = 1.$$

Define, for each real number $\alpha$, a measure $P_{\alpha}$, such that $X$ under $P_{\alpha}$ solves the equation $$\mathrm dX_t = \alpha X_t\,\mathrm dt + \sigma X_t\,\mathrm dB^{\alpha},$$

where $B^{\alpha}$ is a Brownian motion under $P_{\alpha}$. Give an explicit expression for the Radon-Nikodym derivative (likelihood process) $$L^{\alpha} = \frac{\mathrm dP_{\alpha}}{\mathrm dP_0},$$ on $\mathcal{F}_t$.

So this is the question. And I guess you're supposed to use the Itô formula. But I've had a hard time grasping the question. Some guidance on how I could think and where I should begin would be more than appreciated!

(This is my first post on this site and also the first time i use TeX so might not look very good, hopefully you'll understand anyway!)

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Yep, looked like my attempt of using TeX was not very good. –  Good guy Mike Nov 29 '12 at 20:21