I am trying to understand the Radon Nikodym derivative. My professor often writes the measure change from $\mathbb{P}$ to $\mathbb{Q}$ as: $$\eta_t=\dfrac{d\mathbb{Q}}{d\mathbb{P}}|_{\mathscr{F}_t}$$ But then he might write the measure change from $\mathbb{Q}$ to $\mathbb{Q^*}$ as $$\dfrac{d\mathbb{Q^*}}{d\mathbb{Q}}|_{\mathscr{F}_T}$$ What does the notation "given" ${\mathscr{F}_T}$ mean here? In one part of my lecture notes, he writes: $$\dfrac{d\mathbb{Q^*}}{d\mathbb{Q}}|_{\mathscr{F}_T} = \dfrac{e^{-r_T}}{E_0[e^{-r_T}]}$$ Does $E_0[e^{-r_T}] = E[e^{-r_T}|\mathscr{F}_0]$? What meaning does this have? Using the definition of $r_t$ (which I have left out purposefully since it adds no value to the question, one can easily calculate: $$\tilde{\eta_T}:=\dfrac{d\mathbb{Q^*}}{d\mathbb{Q}}|_{\mathscr{F}_T}$$ where $k \in \mathbb{R}$. Does this equation hold for every $t \leq T$? In particular, does $$\tilde{\eta_t}:=\dfrac{d\mathbb{Q^*}}{d\mathbb{Q}}|_{\mathscr{F}_t}?$$ I suppose my question may be like a notation one, but I am hoping this notation is well known enough for someone to explain it to me here. To paraphrase my questions: What meaning does $E_0[]$ have? What does it mean to calculate the Radon Nikodym derivative with respect to a filtration at time $t$? How can I better understand this notation?
• This asks about a notational issue, addressed in the answer below, then about at least two mathematical ones, impossible to answer without the specifics of the situation. Note however that the $d$ part of $dW_t^{Q^*}$ and $dW_T^{Q^*}$ might be a typo. – Did Nov 9 '15 at 7:25
The notation $|_{{\mathscr F}_T}$ means that one is considering the restrictions of the measures in question to the $\sigma$-algebra $\mathscr F_T$.