Formula (37.2) in ' On QF' presents SDE for bond price
[$] dZ ( s , T ) = \mu ( s , T ) Z ( s , T ) ds + \sigma ( s , T ) Z ( s , T ) dX ( s ) [$] (1)
Let coefficients be deterministic and continuous functions on [ 0 , T ].
It is clear that sde should be interpreted in integral form. How does it should be integrated over [ 0 , t ] or over [ t , T ] ?
We know that eq (1) has sense only as integral eq
[$]Z ( t , T ) - Z ( s , T ) = \int_s^t \mu ( u , T ) Z ( u , T ) du + \int _s^t \sigma ( u , T ) Z ( u , T ) d X ( u ) [$]
Now we should specify the spot and future dates for dates s and t. Let s be spot and t is a future date. Then putting for example t = T we should guarantee that
1)
Z ( s , T ) < 1 for s < T ,
2) Z ( T , T ) = 1 , and 3) it is not clear why spot price Z ( s , T ) at is random while it is known at s. I think 1-2 problems. probably could be overcome. But 3) is needed to be explained.
Other case when t is spot and s is past date we have similar problems.Unfortunately these technical problems did not discuss in the book but they are effect their practical applications too. What we should conclude in a case when we say that we have a good approximation historical data with model (1) and an opponent say that there is 10% a chance that Z ( t , T ) > 1?