This question is in relation to the vasicek model, where i am trying to find the solution.

I have this term: $-\int_{t}^{T} \sigma \int_{t}^{s} e^{-\kappa(s-u)} d W(u) d s$

I need to change the integrals like this: $-\sigma \int_{t}^{T} \int_{u}^{T} e^{-\kappa(s-u)} d s d W(u)$

but i have absolutely no idea how he goes from $\int_{t}^{s}$ to $\int_{u}^{T}$.

I know about the changing order of integration, but this is (from what i understand) not what has been done. The rest of the integration should be straight forward, once i understand how i do this.

  • 1
    $\begingroup$ The double integral is on $(u, s) \in [t, T]^2$ with $u<s$, so: $$\int_t^T \int_t^s f(u,s) dW(u) ds = \int_t^T \int_t^T 1_{u<s} f(u,s) dW(u) ds = \int_t^T \int_t^T 1_{u<s} f(u,s) ds dW(u) = \int_u^T \int_t^T f(u,s) ds dW(u) $$ Of course, the function $f$ has to satisfy some conditions so that all the terms above have a meaning, in particular $\int_u^T (\int_t^T f(u,s) ds)^2 du < \infty$ which is the case here. $\endgroup$
    – byouness
    May 14 at 14:52
  • $\begingroup$ Hi: Someone else hopefully can explain it in more detail but the idea is that $u$ has to be less than $s$ ( e has to be raised to a negative power ) in order to have mean reversion. But in the original integral, $s$ goes from $t$ to $T$ so the bottom limit in the outside integral has to be changed to be from $t$ to $u$ for that condition to hold. Also, dW(u) ( which can be thought of as W(du) ) goes from $t$ to $s$ in the original integral, so the upper limit in the inside integral needs to be changed from $s$ to $T$. Then, you can flip the order of the integrals. $\endgroup$
    – mark leeds
    May 14 at 15:10
  • $\begingroup$ @mbih: byouness explained it in a much clearer way than I did but the idea is the same. $\endgroup$
    – mark leeds
    May 14 at 15:11
  • $\begingroup$ @markleeds what is this called? Im not getting what you guys are exlaining to me, even though it sounds totally correct. There must be some way i can read about this concept, but i dont know the name / method. $\endgroup$
    – mbih
    May 15 at 16:12
  • $\begingroup$ It's a form of Fubini's theorem that permits to change the order of integrals in a double integral. $\endgroup$
    – byouness
    May 15 at 19:40