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Is this correct?
I am trying to integrate a Brownian motion where the time-variable itself is a function of time, and I was wondering whether the following is correct, i.e. can you think of the t in W_t as a normal variable.This comes up in trying to get from the Vasicek/HW model to the HJM model where one has to integrateThere might be another way to do the above integral, if there is, any details would be most welcome.Thanks in advance.
Is this correct?
Your integral is not an Ito integral, it is a path-by-path integral, i.e. for each w in your sample space you are computing just a plain 'ol integral of a continuous function, so I see no problem - the only care you may have to take with this is remember you are taking the integral of a non-differentiable function. And I think in the case of you can do a u-sub u = f(s) even though g is not differentiable I think.