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What is the right approach to take and find the moments of the following: $$\mathcal{Z}_t=\int\mathcal{W}_t^k\,d\mathcal{W}_t=?$$ $$\mathcal{W}_t \sim \mathcal{N}(0, t),\ k=2,3...$$ $$\operatorname{E}(\mathcal{Z}_t)=?$$ $$\operatorname{Var}(\mathcal{Z}_t)=?$$

I know that $\mathcal{Y}_t=\int\mathcal{W}_t\,d\mathcal{W}_t=\frac{\mathcal{W}_t^2-t}{2}$ and $\operatorname{E}(\mathcal{Y}_t)=0$. One can derive it from Ito's lemma and the fact that $\operatorname{Var}(\mathcal{W}_t)=t=\operatorname{E}(\mathcal{W}_t^2)$. Is there another way to prove it?

Is there also some generic rules to deal with $\int t\,d\mathcal{W}_t$ and $\int \mathcal{W}_t\,dt$?

Thanks!

  • 2
    Hint: A stochastic integral is a martingale that starts at zero. What does that say about its mean?2011-04-01
  • 0
    @byron-schmuland Thanks! Yes, it is always zero.2011-04-04

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