35. Brownian Motion
本附录章为指针章 (pointer chapter)：原讲义只写一句「详见 He (2019d)（随机微积分笔记）」。为便于后文 [[stochastic-integral|第 36 章随机积分]]、[[itos-lemma|第 37 章 Itô 引理]] 的阅读，下面给出标准布朗运动 (standard Brownian motion) 的简要回顾——这是后续所有连续时间资产定价记号的基础。
This appendix chapter is a pointer chapter: the original lecture notes only say "See He (2019d) (Stochastic Calculus Notes) for details." To support the reading of [[stochastic-integral|Ch 36 Stochastic Integral]] and [[itos-lemma|Ch 37 Ito's Lemma]] below, here is a brief recap of standard Brownian motion — the foundation of all the continuous-time asset-pricing notation that follows.
35.1 Standard Brownian Motion (Recap)
一个随机过程 \(\{Z_t\}_{t\ge0}\) 称为标准布朗运动 (standard Brownian motion / Wiener process)，若满足：
- \(Z_0=0\)；
- 独立增量 (independent increments)：对 \(0\le s
- 高斯增量 (Gaussian increments)：\(Z_t-Z_s\sim\mathcal N(0,\,t-s)\)；
- 样本路径连续（但几乎处处不可微）。
由此立即得到后文反复使用的微分启发式记号：在 \([t,t+dt]\) 上的增量
A stochastic process \(\{Z_t\}_{t\ge0}\) is a standard Brownian motion (Wiener process) if:
- \(Z_0=0\);
- independent increments: for \(0\le s
- Gaussian increments: \(Z_t-Z_s\sim\mathcal N(0,\,t-s)\);
- sample paths are continuous (but almost-everywhere non-differentiable).
This immediately gives the differential heuristic notation used repeatedly below: the increment over \([t,t+dt]\)
$$dZ_t\sim\mathcal N(0,\,dt),\qquad \mathbb E[dZ_t]=0,\qquad \mathbb E[(dZ_t)^2]=dt$$
正是这条 \(\mathbb E[(dZ_t)^2]=dt\)（而非 \(0\)）使得二阶项在 Itô 微积分中不可忽略——见 [[itos-lemma|第 37 章 Lemma 37.1]] 的微积分法则。严格构造与正则性条件见 He (2019d)。
It is precisely this \(\mathbb E[(dZ_t)^2]=dt\) (not \(0\)) that makes the second-order term non-negligible in Itô calculus — see the calculus rules in [[itos-lemma|Ch 37 Lemma 37.1]]. For the rigorous construction and regularity conditions, see He (2019d).
References
- He, X. (2019d). Stochastic Calculus Notes by Xindi He.
- He, X. (2020–2024). Asset Pricing (lecture notes), Ch. 35.