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Can anyone help me with the following problem?

Let $W(t), t\geq 0$ be a Brownian motion with filtration:$F(t)$. Let $0\leq s\leq t$.

1- Show that $E\left [ W^{3}(t)\mid F(s) \right ]=W^{3}(s)+3(t-s)W(s)$. where $E\left [ (*)\mid* \right ]$ is the conditional expectation.

2- Show that: $ W^{3}(t)-3\int_{0}^{t}W(u)du$ is a Martingale.

For the First Part: I tried the following: $E\left [ W^{3}(t)\mid F(s) \right ]=E\left [ W^{3}(t)-W^{3}(s)+W^{3}(s) \right ]=E\left [ W^{3}(t)-W^{3}(s) \right ]+E\left [W^{3}(s) \right ]=E\left [ (W(s)-W(t))^{3} +3W(t)W(s)(W(t)-W(s))\right ]+W^{3}(s)=...$

I couldn't move from that point. Can anyone write in detail how I can complete the solution.

For the second part: I did the following $E\left [ W^{3}(t)-3\int_{0}^{t}W(u)du\mid F(s) \right ]=E\left [ W^{3}(t) \mid F(s)\right ]-3E\left [ \int_{0}^{t}W(u)du\mid F(s) \right ]=W^{3}(s)+3(t-s)W(s)-3\int_{0}^{t}E\left [ W(t)\mid F(s) \right ]du$.

Can anyone let me know how to finish my proof?

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What do you know? The definition of conditioning ? Itô's formula? –  Did Mar 22 '12 at 6:39
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1 Answer 1

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Note that $W(t)^3=W(s)^3+3W(s)^2V+3W(s)V^2+V^3$ with $V=W(t)-W(s)$. Hence, $\mathrm E(W(t)\mid F(s))=U_3+3U_2+3U_1+U_0$ with $U_i=\mathrm E(W(s)^iV^{3-i}\mid F(s))$.

To compute each $U_i$, one can apply the principle David Williams calls:

Leave out everything that is measurable, integrate everything that is independent.

Here, $W(s)$ is measurable with respect to $F(s)$ and $V$ is independent on $F(s)$ hence $U_i=W(s)^i\mathrm E(V^{3-i})$ for every $i$. Furthermore, the distribution of $V$ is symmetric hence $\mathrm E(V)=\mathrm E(V^3)=0$, and $V$ is gaussian with variance $t-s$ hence $\mathrm E(V^2)=t-s$. Putting everything together, one gets $$ \mathrm E(W(t)\mid F(s))=W(s)^3+3W(s)^2\cdot0+3W(s)\cdot(t-s)+0, $$ that is, $$ \mathrm E(W(t)\mid F(s))=W(s)^3+3(t-s)\cdot W(s). $$ As regards 2., thanks to 1., this reduces to the fact that $$ (t-s)W(s)=\mathrm E\left(X_{s,t}\big\vert F(s)\right),\quad X_{s,t}=\int_s^tW(u)\mathrm du.\tag{E} $$ Do you see why (E) and 1. together imply 2.? Do you see why (E) holds?

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