joint distribution of $(W(1),W(3),W(3)-W(2))$ for a brownian motion $(W(t))_{t \geq 0}$

Question

Let $(\Omega,\mathcal{F},P)$ be a probability space, $(W(t),t \ge 0)$ a Brownian motion and $(\mathcal{F}_t,t \ge 0)$ its natural filtration.

What is the joint probability distribution of $(W(1),W(3),W(3)-W(2))$?

All I know is that $W(1) \sim \mathcal{N}(0,1),W(3) \sim \mathcal{N}(0,3), W(3)-W(2) \sim \mathcal{N}(0,1)$ and that $W(3)-W(2)$ is independent of $W(1)$.

A hint would be great. Merci!

Answer 1

Hints:

1. Since $(W_t)_{t \geq 0}$ is a Brownian motion, it has independent normally distributed increments; in particular, $W_3-W_2, W_2-W_1,W_1$ are independent random variables which are Gaussian with mean $0$ and variance $1$. This implies that $(W_1, W_2-W_1,W_3-W_2)$ is (jointly) Gaussian with mean $\mu=0$ and covariance matrix $$C = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}.$$
2. Recall the following statement: Let $X: \Omega \to \mathbb{R}^n$ be a Gaussian random vector $X \sim N(\mu,C)$ and $A \in \mathbb{R}^{n \times n}$. Then $Y:=AX$ is Gaussiwn with mean $A \mu$ and covariance matrix $A C A^T$.
3. Show that $Y= (W_1,W_3,W_3-W_2)$ satisfies $$Y = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} W_1 \\ W_2-W_1 \\ W_3-W_2 \end{pmatrix}.$$
4. Conclude.