A standard <Brownian motion> is a real process $(B_t)_{t\geq0}$ with $B_0=0$ almost surely, <continuous> paths, <independent> increments, and $B_t-B_s\sim N(0,t-s)$ for $0\leq s<t$. If a filtration is specified, require adaptedness and independence of each increment from the past sigma-algebra. Thus \b[the increments are centered Gaussian with <variance> equal to elapsed time].

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 27J / a / Solution

 Solution (source code)