Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 27K / i

A standard Brownian motion has $B_0=0$, continuous paths almost surely, and independent increments with $B_t-B_s\sim N(0,t-s)$ for $0\le s<t$. With a specified filtration, the increment is also independent of the information available at time $s$.