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I have a question regarding integration order reversal in a stochastic integral. This is a homework problem of the form "Show this is true". My problem is 1) my results are not exactly the same as the results in the book and 2) there is an equation in the book that does resolve the issue but I am concerned it involves circular logic, since we are trying to justify that equation.

The text is Shreve's "Stochastic Calculus for Finance 2", question 10.8, part c, page 458.

The question is prove these two quantities are the same.

Quantity 1:

$$ d\left( - \int_t^T f(0,v) \, dv - \int_0^t \hat{\alpha} (u,t,T) \, du - \int_0^t \hat{\sigma} (u,t,T) \, dW(u) \right) $$

Quantity 2:

$$ \left( f(0,t) + \int_0^t \alpha(u,t) \, du + \int_0^t \sigma(u,t) \, dW(u) \right) dt - \int_t^T \alpha(t,v) \, dv \, dt - \int_t^T \sigma(t,v) \, dv \, dW(t)$$

Definition of $\hat{\alpha}$ and $\hat{\sigma}$:

$$\hat{\sigma} (u,t,T) = \int_t^T \sigma(u,v) \, dv$$

$$\hat{\alpha} (u,t,T) = \int_t^T \alpha(u,v) \, dv$$

Here is my differential of quantity 1: $$ f(0,t) dt - \int_t^T \alpha(t,v) \, dv \, dt - \int_t^T \sigma(t,v) \, dv \, dW(t) + \int_0^t \alpha(u,t) \, dt \, du + \int_0^t \sigma(u,t) \, dt \, dW(u)$$

As you can see, almost everything is the same except the following:

$$ \int_0^t \alpha(u,t) \, dt \, du + \int_0^t \sigma(u,t) \, dt \, dW(u) \, \, ?? \, \, \left(\int_0^t \alpha(u,t) \, du + \int_0^t \sigma(u,t) \, dW(u)\right) dt$$

The 2 questions marks in the middle indicates possible equality. In fact, there is an equation in the book that does show that these two quantities are the same. However, I am concerned if the problem question wants us to prove the invariance of the order of integration, perhaps I did the differentials incorrectly and the 2 equations are the same without needing to go to the fix-it equation.

Thanks

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1 Answer

The solution to your problem is a theorem usually called stochastic Fubini theorem. It should be included in any textbook covering HJM. I think Shreve leaves it as homework, doesn't he? Different version with different conditions exist. This allows you to revise order of integration with the stochastic integral.

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