Brownian Motion Product Moments at times $0 < u < u + v < u + v + w,$ where $u, v, w > 0$

Question

The following is from Pinsky & Karlin's $\textit{Introduction to Stochastic Modelling}$:

Consider a standard Brownian motion $\{B(t); t ≥ 0\}$ at times $0 < u < u + v < u + v + w,$ where $u, v, w > 0.$

(a) Evaluate the product moment $\mathbb{E}[B(u)B(u + v)B(u + v + w)]$.

(b) Evaluate the product moment $\mathbb{E}[B(u)B(u + v)B(u + v + w)B(u + v + w + x)]$ where $x > 0$.

I tried to approach this question by separating the Brownian motions: $$B(u+v) = B(u) + (B(u+v) - B(u)) $$and$$ B(u+v+w) = B(u) + (B(u+v) - B(u)) + (B(u+v+w) - B(u+v))$$ and then substituting them back into the expectation.

From which I obtained the following:

$E[B(u) B(u+v) B(u+v+w)] = E[ B(u) (B(u) + (B(u+v) - B(u))) (B(u)$

$\hspace{6.3cm}$$ + (B(u+v) - B(u)) + (B(u+v+w) - B(u+v))) ]$

But I am stuck at this point.. How can I proceed from here for part (a)? Also, for part (b), is the approach the same as above? Thanks and some clarifications will be great.

Answer 1

Hints:

1. Recall that the following processes are martingales: $B_t$, $B_t^2-t$, $B_t^3-3tB_t$.
2. Deduce from the tower property that $$\mathbb{E}(B(u)B(u+v)B(u+v+w)) = \mathbb{E}(B(u) B(u+v) \underbrace{\mathbb{E}(B(u+v+w) \mid \mathcal{F}_{u+v})}_{B(u+v)}).$$
3. Another application of the tower property yields $$\mathbb{E}(B(u)B(u+v)B(u+v+w)) = \mathbb{E}(B(u) \underbrace{\mathbb{E}(B(u+v)^2 \mid \mathcal{F}_u)}_{B_u^2-u^2+(u+v)^2})= \mathbb{E}(B_u^3)=0$$ which solves (a).
4. For (b) note that (using exactly the same resoning as above) $$\mathbb{E} (B(u)B(u+v)B(u+v+w)B(u+v+w+x)) = \mathbb{E} \bigg[ B(u) \bigg( B(u+v)^3-B(u+v) ((u+v+w)^2-(u+v)^2) \bigg) \bigg].$$ Use the tower property to condition on $\mathcal{F}_u$ and use that $B_t^3-3tB_t$ and $B_t$ are martingales.