Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 201 / 4 / a / Solution

A standard real Brownian motion relative to a filtration $({\mathcal{F}}_t)$ is an adapted stochastic process with $B_0=0$ almost surely, continuous paths almost surely, andEquivalently, it has independent increments and stationary increments, each with a normal distribution of mean zero and variance equal to the elapsed time, together with continuous paths and the stated initial value. The filtration may be its natural filtration. A Brownian motion started at $z$ is $z+W_t$ for a standard process $W$; multidimensional standard Brownian motion has independent standard real coordinate processes. The normalization is variance $t$ at time $t$.