Bayes rule is an important theory in statistics. In this following, we discuss one of the presentation, which can be applied to interest rate model, see [MR05].
Proposition 1 (Bayes rule) Let be two probability measures on the probability space , satisfying Let be a filtration with . Then,
Proof: Note that and are also probability measures on , we denote the restricted probabilities by and respectively. Then, are martingale, i.e. , see Lemma 1 of here.
It’s enough to show,
It follows by
In particular, with in the above proposition, we have
In this below, we consider an application of Bayes rule to the interest rate theory. Suppose is the short rate, and is the unique equivalent martingale measure. Then, the price at time of the derivative with maturity and payoff measurable is written by
In particular, the price of zero bond is, with ,
Then, by the Bayes rule, the derivative price of (1) can be written w.r.t. forward measure:
Proposition 2 For all , the process
is an -martingale under .
Proof: It’s enough to show
which is indeed a consequence of (3) with .