Hello,

I'm struggling with an application of bayes rule in a proof and I hope you can help me.

At first some preliminaries:

**P(t,T) is the price of a Zero-Coupon-Bond and B(t) is a discount factor<<--I think this isnt necessary to know in order to understand the problem below. But just for the completeness.

We have a prop measure Q and define an equivalent prop. measure $\displaystyle Q^T$by

$\displaystyle \frac{dQ^T}{dQ}=(P(0,T)B(T))^{-1}$

on a Filtration F_T, generated by Brownian Motions.

In my context we have in particulary for $\displaystyle t\leq T$

$\displaystyle \frac{dQ^T}{dQ}|F_t=\frac{P(t,T)}{P(0,T)B(t)}$

Problem:

Now one can show, that $\displaystyle \frac{P(t,S)}{P(t,T)}$ is a martingale.

But I do not understand the key step in the argumentation where bayes rule is used. The argumentation is for $\displaystyle u\leq t\leq T\wedge S$

$\displaystyle \mathbb{E}_{Q^T}\bigg[\frac{P(t,S)}{P(t,T)}\bigg| \mathcal{F}_u\bigg]=\frac{\mathbb{E}_\mathbb{Q}[\frac{P(t,T)}{P(0,T)B(t)}\frac{P(t,S)}{P(t,T)}| \mathcal{F}_u]}{\frac{P(u,T)}{P(0,T)B(u)}} = \frac{\frac{P(u,S)}{B(u)}}{\frac{P(u,T)}{B(u)}}= \frac{P(u,S)}{P(u,T)}$

The first equation is justified by bayes rule. But how?

I think it should be something like:

Represent the left Expectation through the Radon-Nikodym-Derivative and then use bayes, but I just can't complete it.

P.s. If some more "context" is needed, just say. But, I think the problem above is just a pure stochastic question.