am trying to find the expectation of $\int_0^t W_s ds$, with $W_s$ being the Standard Wiener process. I am trying to use Ito's formula, by decomposing as:

$$ \frac{W_t^3}{6} = \frac{1}{2}\int_0^t B_s^2 dB_s + \int_0^t B_s ds $$

Then, applying the expectation on both sides, I get that:

$$ E\left[\int_0^t B_s ds\right]= - E\left[\frac{1}{2}\int_0^t B_s^2 dB_s\right] $$

In this case, I am wondering if I am allowed to use Fubini's theorem to move the expectation on the right into the integral. Would it be defined for the $dB_s$ integrand? Thanks.

  • 1
    $\begingroup$ Fubini indeed: $$E\left(\int_0^t W_s ds\right)=\int_0^t E(W_s) ds=\int_0^t 0 ds=\ldots$$ $\endgroup$ – Did Apr 28 '16 at 6:58

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