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How does one compute $E[(\int_0^T W_s ds)^2]$ where $(W_t)_{t \in [0,T]}$ is standard Brownian motion in $(\Omega, \mathscr F, \mathbb P)$?

Apparently proving

$$\int_0^T W_s ds = \int_0^T (T-s) dW_s \tag{*}$$

might need to assume that $E[(\int_0^T W_s ds)^2] < \infty$


I think I can just use Ito's lemma to show $(*)$ and then use Ito isometry to show that

$$E[(\int_0^T W_s ds)^2] = E[(\int_0^T (T-s) dW_s)^2] = (\int_0^T (T-s)^2 ds) < \infty$$

Is that okay? If not, why?

Regardless, I'm still curious as to how compute $E[(\int_0^T W_s ds)^2]$ besides using the representation if it's possible? I don't think we encountered $\int_0^T W_s ds$ in classes

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Assuming that $E[W_tW_s]=\sigma^2\min(t,s)$, \begin{align} E\left[\left(\int_0^TW_s\,\mathrm ds\right)^2\right]&=E\left[\int_0^TW_t\,\mathrm dt\int_0^TW_s\,\mathrm ds\right]\\ &=\int_0^T\int_0^T\sigma^2\min(t,s)\,\mathrm dt\,\mathrm ds\\ &= \int_0^T\int_0^s\sigma^2 t\,\mathrm dt\,\mathrm ds + \int_0^T\int_s^T\sigma^2 s\,\mathrm dt\,\mathrm ds\\ &=\int_0^T\sigma^2\frac{s^2}{2}\,\mathrm ds + \int_0^T\sigma^2(T-s)s\,\mathrm ds\\ &= \sigma^2\left(\frac{T^3}{6}+\frac{T^3}{2}-\frac{T^3}{3}\right)\\ &= \sigma^2\frac{T^3}{3} \end{align}

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  • $\begingroup$ Thanks Dilip Sarwate. I was afraid of going through using a definition with partitions or something. Hahaha. Does Fubini's indeed hold for switching an expectation and a double integral? Well O guess if we can switch for two integrals we can switch for n integrals (n=3). Just haven't seen that before $\endgroup$ – BCLC Dec 24 '15 at 5:33
  • $\begingroup$ Also why 'assuming that' rather than 'since' ? $\endgroup$ – BCLC Dec 24 '15 at 5:39
  • $\begingroup$ "Also why 'assuming that' rather than 'since' ?" Because I have no idea as to what the most popular definition of standard Brownian motion is? $\endgroup$ – Dilip Sarwate Dec 24 '15 at 14:24
  • $\begingroup$ T^3 not T^2?... $\endgroup$ – BCLC Dec 26 '15 at 19:35
  • $\begingroup$ Dilip Sarwate, I think you can remove the question marks $\endgroup$ – BCLC Dec 28 '15 at 10:02

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