Definition
Standard Brownian Motion
A standard Brownian motion (or Wiener process) is a stochastic process $\{B_t\}_{t \geq 0}$ (Definition~Stochastic Process) on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ (Definition~Probability Space) satisfying:
- Initial value: $B_0 = 0$ $\mathbb{P}$-a.s.
- Independent increments: For $0 \leq t_0 < t_1 < \cdots < t_n$, the increments $B_{t_1} - B_{t_0}, \ldots, B_{t_n} - B_{t_{n-1}}$ are independent (Definition~Independence).
- Stationary Gaussian increments: For $s < t$, $B_t - B_s \sim \mathcal{N}(0, t-s)$.
- Continuous paths: $t \mapsto B_t(\omega)$ is continuous $\mathbb{P}$-a.s.
Dependency Graph
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