Hi,

I was reading Shreve Vol 2 and had a question on application on Ito's Lemma for Vasicek Interest rate model in Chapter 4.

While taking partial differential with respect to t for the function f(t,X(t)-the example treats X(t) as a constant. My question was that shouldn't X(t) -since its a function of time should also be treated as a variable with respect to time and accordingly differentiated? Pardon me if this is a very basic question.


Thanks,
Atul

For those that do not own Shreve, maybe you can post the equation/your question in LaTex here.

yes, sure. I do not have Latex installed so would try and write the equations. The solution of Vasicek SDE is :

R(t)= e^-bt.R(0)+a. (1-e^-bt)/b+ sigma.e^-bt.X(t)
where X(t) is an ito integral from 0 to t with a time deterministic integrand. Shreve defines f(t,x) as:
 e^-bt.R(0)+a. (1-e^-bt)/b+ sigma.e^-bt.x
and then calculates:
f_t = -b.e^-bt.R(0)+ ae^-bt-sigma.b.e^-bt.x
My question is that x should also be differentiated with respect to t since X is a function of t. But the book doesn't seem to do that. 
Please let me know in case my equations are not clear.

yes, sure. I do not have Latex installed so would try and write the equations. The solution of Vasicek SDE is :

R(t)= e^-bt.R(0)+a. (1-e^-bt)/b+ sigma.e^-bt.X(t)
where X(t) is an ito integral from 0 to t with a time deterministic integrand. Shreve defines f(t,x) as:
 e^-bt.R(0)+a. (1-e^-bt)/b+ sigma.e^-bt.x
and then calculates:
f_t = -b.e^-bt.R(0)+ ae^-bt-sigma.b.e^-bt.x
My question is that x should also be differentiated with respect to t since X is a function of t. But the book doesn't seem to do that. 
Please let me know in case my equations are not clear.

To get a proof we start with a finite difference approximation. Common term is 
f ( t + h , X ( t + h ))  -  f ( t , X ( t ))  =  [$]I_1  +  I_2[$]  
where 
 [$]I_1[$]  =   f ( t + h , X ( t + h ))  -  f ( t + h , X ( t )) . We need twice continuous differentiability with respect to X
 [$] I_2[$]   =  f ( t + h , X ( t ))  -  f ( t , X ( t ))  . We need 1 time continuous differentiability with respect to t for any fixed X
Then we use Taylor's formula and sum up all  differences

Yes, but when expanding the Taylor series and taking partial differential with respect to t-would you treat X as a constant? If yes, please let me know why is that the case.

Yes, but when expanding the Taylor series and taking partial differential with respect to t-would you treat X as a constant? If yes, please let me know why is that the case.

Yes X ( t ) does not change when we consider the difference f ( t + h , X ( t ))  -  f ( t , X ( t )) as a function of h

It's not that X is a "constant," it's that it's not really a function of time in the usual sense of what a "function" is.