Consider two Wiener processes: $$ \begin{aligned} X_a &\sim\mathcal N(0,a) \\ X_{a-b} &\sim\mathcal N(0,a-b) \end{aligned} $$ How do I show that: $$ X_a - X_{a-b} \sim\mathcal N(.,.) $$ That is, how do I show that $X_a - X_{a-b}$ is Gaussian?
I know that $E(X_a - X_{a-b}) = 0$
and $$ \begin{aligned} \text{Var}(X_a - X_{a-b}) &= E(X_a^2) - E(X_{a-b}^2) \\ &= \text{min}(a,a) -\text{min}(a-b,a-b) \\ &= a - a - b \\ &= b \end{aligned} $$
Note that both $X_a$ and $X_{a-b}$ have the following properties:
- They are normally distributed with mean $0$.
- Their sample paths are continuous.
- Its covariance function is $\rho(s,t) := \text{Cov}(X_s,X_t)=E(X_sX_t)=\text{min}(s,t)$.