Vector Brownian Motion

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $m\ge1$ be a natural number.

An $m$-dimensional Brownian motion on $(\Omega,\mathcal{F},P)$ is a family $W=(W^{1},\dots,W^{m})$ of stochastic processes $W^{j}=(W^{j}_t)_{t\ge0}$ on $(\Omega,\mathcal{F},P)$ such that:

(i) each $W^{j}$ ($1\le j\le m$) is a standard Brownian motion;

(ii) the generated $\sigma$-algebras $\sigma(W^{j}_t:t\ge0)$ ($1\le j\le m$) are independent (a condition that is vacuous when $m=1$).

For $t\ge0$ we write $W_t$ for the tuple $(W^{1}_t,\dots,W^{m}_t)$, a point of the Euclidean space $\mathbb{R}^{m}$, and the processes $W^{1},\dots,W^{m}$ are called the components of $W$.