A process $(X_t)_{t\geq0}$ is <Brownian motion> in $\mathbb R^d$ when $X_0=0$, its paths are almost surely continuous, and for $0\leq t_0<\cdots<t_k$ the increments $X_{t_j}-X_{t_{j-1}}$ are independent centered <multivariate normal distribution>[Gaussian vectors] with covariance $(t_j-t_{j-1})I_d$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 201 / 4 / a / Solution

 Solution (source code)