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Let $(W_t)$ a Brownian motion.

  1. Express $W_t^8$ as a sum of Itô and Riemann integral.
  2. Use part 1) to deduce $\mathbb E(Z^8)$ where Z has an $\mathcal N(0,1)$ distribution.

This is a financial mathematics question, I've reviewed the lecture and think it for about two hours, I don't really have any ideas. Is there anyone who could give me some help?

Ryan
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    Do you know Ito's formula? Basically, if say the question was $W_t^2$ was instead of $W_t^8$, how would you do it? – Sarvesh Ravichandran Iyer Nov 01 '20 at 15:38
  • take integral of ∆s respect to Ws from 2 to t? – Ryan Nov 01 '20 at 15:46
  • Where $\Delta$s mean what? (I.e. what is their definition?) Either way, you need to think about what you'd like to integrate with respect to $W_s$ (i.e. the Ito part) to get $W_t^8$. Take inspiration from the $2$ case : maybe try to write it for $3,$ then $4$ and get a pattern. – Sarvesh Ravichandran Iyer Nov 01 '20 at 15:49
  • The integral should be respected to dWs or dS? – Ryan Nov 01 '20 at 16:04
  • There's a part that is with respect to $dW_s$ (the Ito part) and a part with respect to $ds$ (the Riemann part). Now, from the $W_t^2$ case, for which you think you have seen the answer, I want you to take a guess of the $W_t^8$ case, along with how you would want to argue it. If you don't know the $W_t^2$ case, then I want to know, for example which functions of $W_t$ have you computed the integral w.r.t $dW_s$ of (i.e. I want to know how comfortable you are with these calculations) – Sarvesh Ravichandran Iyer Nov 01 '20 at 16:10
  • Integral from 0 to 8 Δt μ St dt + Integral from 0 to 8 Δt σ St dWt – Ryan Nov 01 '20 at 16:15
  • Does this answer your question? FInancial Itô integral question – callculus42 Nov 01 '20 at 16:55
  • No actually, that question was closed for some reasons – Ryan Nov 01 '20 at 16:58
  • Ah yes, back here. Now, I need to be able to understand your language. Do you have your lecture notes? Or which textbook is your professor following? Please attach anything that reflects what notation you are using and what you know so far. – Sarvesh Ravichandran Iyer Nov 02 '20 at 07:01

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