Standard Brownian Motion

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $\mathbb{R}$ be the real numbers. A stochastic process $(B_t)_{t\ge0}$ on $(\Omega,\mathcal{F},P)$, indexed by the nonnegative real numbers, is a standard Brownian motion if:

(i) (Initial value) $B_0=0$ almost surely (the set $\{B_0=0\}=B_0^{-1}(\{0\})$ is an event, since $B_0$ is a random variable and $\{0\}$ is a Borel set);

(ii) (Almost surely continuous paths) almost surely, the path

is continuous at every point of $[0,\infty)$ relative to $[0,\infty)$, both $[0,\infty)$ and the codomain $\mathbb{R}$ carrying the metric of the real line;

(iii) (Gaussian increments) for all real $0\le s<t$, the increment $B_t-B_s$ is a Gaussian random variable whose expectation and variance — defined and finite by Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector — satisfy

(iv) (Independent increments) $(B_t)_{t\ge0}$ has independent increments: for every natural number $p$ and all real numbers $0\le t_0<t_1<\dots<t_p$, the increments

are independent.