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I am asked to find $Var(W_t^3- \int_0^t3W_sds)$. This is what I have done so far:

$VarY_t=\mathbb{E}(Y_t)^2=\mathbb{E}(W_t^6-2W_t^3\int_0^t3W_sds+(\int_0^t3W_sds)^2)$

I calculated by applying Ito formula twice on $W_t^6$ that $\mathbb{E}(W_t^6)=15t^3$

but I do not know how to calculate the expectation of the last two terms.

T.Sokh
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  • No idea whatsoever to compute $E(W_t^3W_s)$ for $0<s<t$? Come on... – Did Dec 13 '18 at 18:34
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    Have a look here https://quant.stackexchange.com/questions/29504/integral-of-brownian-motion-w-r-t-time – caverac Dec 13 '18 at 18:35
  • isn't that one of those cases where you have to do the increment trick? – Makina Dec 13 '18 at 18:57
  • the third term I understood but for $E(\int_0^tW_t^3W_sds)$ I do not quite get it. Should I rewrite $W_t^3$ sd increments? – T.Sokh Dec 13 '18 at 20:27

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