# Understanding Radon Nikodym derivative

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?

• Have you asked your professor what the notation he uses means?! It is somewhat absurd to ask the world at large if not... – Mariano Suárez-Álvarez Nov 9 '15 at 6:23
• @MarianoSuárez-Alvarez - I will delete the question if this notation is cryptic and uncommon. – Kerry Nov 9 '15 at 6:32
• Why would you delete it? I am merely observing that the very first person you should have asked this question to is precisely your teacher: if he is using the notation, presumably he knows what it means, and if he is using it to communicate with you, then not only you will help yourself by asking him but also help him: no one enjoys talking to someone who has no idea what one is saying! – Mariano Suárez-Álvarez Nov 9 '15 at 6:35
• @MarianoSuárez-Alvarez I had to miss a lecture and I am trying to get caught up before the next one. – Kerry Nov 9 '15 at 6:39
• 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 $\mathscr F_T$.