brownian motion-proof of Martingale 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?
 A: 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?
