This is Girsanov's theorem from wikipedia:
Let $\left\{W_{t}\right\}$ be a Wiener process on the Wiener probability space $\{\Omega, \mathcal{F}, P\}$. Let $\left\{X_{t}\right\}$ be a measurable process adapted to the natural filtration of the Wiener process $\left\{\mathcal{F}_{t}^{W}\right\}$ with $X_{0}=0$. Define the Doléans-Dade exponential $\mathcal{E}(X)_{t}$ of $X$ with respect to $W$ \begin{equation*} \mathcal{E}(X)_{t}=\exp \left(X_{t}-\frac{1}{2}[X]_{t}\right) \end{equation*} where $[X]_{t}$ is the quadratic variation of $X_{t} .$ If $\mathcal{E}(X)_{t}$ is a strictly positive martingale, a probability measure $Q$ can be defined on $\{\Omega, \mathcal{F}\}$ such that we have Radon-Nikodym derivative \begin{equation*} \left.\frac{d Q}{d P}\right|_{\mathcal{F}_{t}}=\mathcal{E}(X)_{t} \end{equation*} Then for each $t$ the measure $Q$ restricted to the unaugmented sigma fields $\mathcal{F}_{t}^{W}$ is equivalent to $P$ restricted to $\mathcal{F}_{t}^{W}$. Furthermore, if $Y$ is a local martingale under $P$, then the process \begin{equation*} \tilde{Y}_{t}=Y_{t}-[Y, X]_{t} \end{equation*} is a Q local martingale on the filtered probability space $\left\{\Omega, \mathcal{F}, Q,\left\{\mathcal{F}_{t}^{W}\right\}\right\}$.
Here are my questions:
- $\mathcal{E}(X)_{t}$ must be a martingale with regard to which filtration ?
- How can a Radon-Nikodym derivative be equal to $\mathcal{E}(X)_{t}$ ? A Radon-Nikodym derivative is a deterministic function but $\mathcal{E}(X)_{t}$ is random (unless $Q$ is a random measure, which is not stated).
- Why do we need $\mathcal{E}(X)_{t}$ to be a strictly positive martingale to be able to define $Q$ ?
- Where can I get a proof of the theorem formulated in this way ? I seem to find a lot of different versions in textbooks, none like the above.