I want to compute $E[∫_0^tB_u \, du ∫_0^sB_u \, du]$ and I know from another source that should be equal to $ts^2/2$. But when I try to compute it like:
$$\begin{align} & E\left[(tB_t- \int_0^tu \, dB_u)(sB_s- \int_0^su \, dB_u)\right] \\ & =E[tsB_t B_s ] – E\left[tB_t \int_0^su \, dB_u\right] – E\left[sB_s \int_0^tu \, dB_u\right] + E\left[\int_0^tu \, dB_u ∫_0^su dB_u\right] \\ & = ts^2 - (ts^2)/2 - s^3/2 + s^3/3\text{ and this is not }ts^2/2. \end{align} $$
I know I took the long way around approach but I still should obtain the same answer.
but I did not check this was indeed the result
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