Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 30 / 3 / a

A scalar process is a Brownian motion in $({\mathcal{F}}_t)$ when it is adapted, has continuous sample paths, starts at zero almost surely, and, for every $0\le s<t$,Independence from the whole past sigma-field is the essential filtration requirement. It implies independent increments over disjoint intervals; the variance $t-s$ fixes the standard Brownian time scale.