
Brownian motion

Check that the process $\displaystyle A_t=B_{t_0+t}B_{t_0}$ satisfies the characteristic properties of Brownian motion: $\displaystyle A_0=0$, $\displaystyle A_t$ is Gaussian with mean 0 and variance $\displaystyle t$, $\displaystyle A$ has stationary independent increments, and is almost surely continuous (maybe you know another characterization of B.M. from your lecture; then adapt the sentence). All of these properties are immediate consequences of the same properties for $\displaystyle B$.