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 ,
The forward measure is the probability defined as
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 .