APPENDIX A:
Proof of Proposition 1. We rewrite (9) as
(A1)
In the general case when the Wiener process has =a and =b, the
distribution of at time is normal given by
(A2)
In our case: , , a =0, b =, , thus (A2) can be expressed as
(A3)
Let . According to the linear transformation rule, is a normal given by
(A4)
Let . By definition, is a lognormal given by . According to the
characterizations of the lognormal distribution, the mean and variance
of are
(A5a)
(A5b)
We have the conditional expectation of the forward rate as
(A6)
Proof of Proposition 2 . Let where is defined above.
According to the linear transformation rule, is a lognormal given by .
The mean and variance of are
(A7a)
(A7b)
On the other hand, according to the characterizations of the lognormal
distribution, the mean and variance of are
(A8a)
(A8b)
Solving the equation (A8a) and (A8b), we get
(A9a)
(A9b)
We know the first negative moment of the lognormal is and have the
conditional expectation of the drift term as
(A10)
where , are given by (A7a) and (A7b).