i am struggeling with the following exercise
Let $r,\mu,\sigma,T>0$ and consider the market model with a money-market account $B$ and one risky asset $S$ such that \begin{eqnarray*} dB(t)&=&rB(t)dt,\quad B(0)=1,\\ dS(t)&=&S(t)(\mu dt+\sigma dW(t)),\quad S(0)>0, \end{eqnarray*} where $(W(t))_{t\ge0}$ is a one dimensional Brownian motion. Let $\gamma$ be progressively measurable and satisfying certrain integrability conditions such that $\mathcal{E}_t(\gamma\circ W)$ is an martingale for $t\le T$ and define $\mathbb{Q}\sim\mathbb{P}$ on $\mathcal{F}_T$ by $d\mathbb{Q}/d\mathbb{P}=\mathcal{E}_T(\gamma\circ W)$, where $\mathcal{E}(W)$ denotes the stochastic exponential of $W$.
(i) Find the Girsanov transformed Brownian motion $W^*$ and the It$\hat o$ decomposition of $\mathcal{S}=S/B$ with respect to $W^*$ for $t\in(0,T]$.
(ii) find $\gamma$ such that $\mathcal{S}$ is a $\mathbb{Q}$-local martingale for $t\le T$.
To part (i): by Girsanov theorem $W^*(t)=W(t) -\int_0^t \gamma(s)ds$ is a Brownian motion under $\mathbb{Q}$. Further we know that $-\gamma=\frac{\mu-r}{\sigma}$ has to hold (uniqueness of representation of an ito process). So is this the answer? or is it possible to calculate the expected value and variance of $W^*(t)$ under $\mathbb{Q}$?
Further can I verify the statement of Girsanov theorem directly? I tried to calculate $\mathbb{Q}(W^*(t)\le u)$ but failed to derive something.
Thanks for your help..