As part of some practice questions for a course I'm taking, I was given the definition of a Brownian motion $W_{t}$ as a unique continous-time stochastic process satisfying:
- $W_{0}=0$
- The function $t \rightarrow W_{t}$ is almost surely everywhere continous.
- The increments $W_{t_{1}} - W_{s_{1}}W_{t_{2}}-W_{s_{2}}$ are independent when $0 \leq s_{1} < t_{1} \leq s_{2}<t_{2}$
- The increment $W_{t} - W_{s}$ has a $N(0, t-s)$ distribution for $0 \leq s < t$.
This seems to make sense and I did some further research into Brownian motions, however then I was asked what the distribution of $W_{1}$ and $(W_{1},W_{2})$ would be.
I don't really know what this means, as I don't know how to express either as any more than some function that is almost everywhere continuous.. it feels like my given definition isn't exactly enough to go off of...