Appendix. Continuous Time

Jun He June 02, 2026

中文English 连续时间Continuous Time 布朗运动Brownian Motion 扩散过程Diffusion Ito引理Ito's Lemma 随机微积分Stochastic Calculus

Appendix. Continuous Time

Note

本章导读本附录(Cochrane 全书收官)是连续时间随机过程机制的简明导论——即如何使用 $dz$ 与 $dt$,支撑全书的连续时间定价(Ch 1.5、17、18、19 等)。连续时间的形式数学颇为吓人(基本随机游走 $z_t$ 不可微),但只需几条直观规则——尤其 $dz^2=dt$ ——就能很快上手。思路与离散时间 ARMA 模型($x_t=\rho x_{t-1}+\varepsilon_t$)类比:先有简单冲击序列(离散 $\varepsilon_t$ ↔ 连续 $dz_t$),再在其上叠加建出更复杂的模型。§A.1 布朗运动;§A.2 扩散模型;§A.3 Ito 引理。

Appendix. Continuous Time

Note

Overview This appendix (the book's closing) is a brief introduction to the mechanics of continuous-time stochastic processes — how to use $dz$ and $dt$ — underpinning the continuous-time pricing throughout the book (Ch 1.5, 17, 18, 19, etc.). The formal mathematics is imposing (the basic random walk $z_t$ is not differentiable), but with a few intuitive rules — especially $dz^2=dt$ — one can use these processes quickly. The ideas are analogous to discrete-time ARMA models ($x_t=\rho x_{t-1}+\varepsilon_t$): start with a simple shock series (discrete $\varepsilon_t$ ↔ continuous $dz_t$), then build more complex models on it. §A.1 Brownian motion; §A.2 diffusion models; §A.3 Ito's lemma.

A.1 布朗运动 / Brownian Motion

基本构件是布朗运动,即离散时间随机游走 $z_t-z_{t-1}=\varepsilon_t$ 的自然推广。随机游走的方差随时间线性增长,故定义布朗运动为满足下式的过程:

A.1 Brownian Motion

The basic building block is a Brownian motion, the natural generalization of a discrete-time random walk $z_t-z_{t-1}=\varepsilon_t$. A random walk's variance scales linearly with time, so a Brownian motion is defined as a process $z_t$ satisfying:

$$z_{t+\Delta}-z_t\sim N(0,\Delta).\tag{A.1}$$

即在通常的随机游走定义上加了正态分布;且非重叠区间的增量相互独立(对应离散时间 $E(\varepsilon_t\varepsilon_{t-1})=0$)。记 $dz_t$ 为任意小时间区间 $\Delta$ 的 $z_{t+\Delta}-z_t$;水平 $z_t$ 是其小增量之和:$z_t-z_0=\int_0^t dz_s$(随机积分)。关键性质:方差随时间线性增长、故标准差随时间的平方根增长,故 $(z_{t+\Delta}-z_t)/\Delta$ 的典型大小为 $1/\sqrt\Delta$——这意味着虽然 $z_t$ 样本路径连续,$z_t$ 却不可微(故 $dz=(dz/dt)dt$ 没有意义)。

This adds a normal distribution to the usual random-walk definition; and increments over non-overlapping intervals are independent (the analogue of discrete-time $E(\varepsilon_t\varepsilon_{t-1})=0$). Write $dz_t$ for $z_{t+\Delta}-z_t$ over an arbitrarily small interval $\Delta$; the level $z_t$ is the sum of its small increments: $z_t-z_0=\int_0^t dz_s$ (a stochastic integral). Key property: the variance scales linearly with time, so the standard deviation scales with the square root of time, and $(z_{t+\Delta}-z_t)/\Delta$ has typical size $1/\sqrt\Delta$ — meaning that although the sample path of $z_t$ is continuous, $z_t$ is not differentiable (so $dz=(dz/dt)dt$ makes no sense).

Important

$dz^2=dt$——不只是期望,而是逐路径成立 / $dz^2=dt$ — not just in expectation, but path by path 由 (A.1) 显然 $\mathbb E_t(dz_t)=0$(注意 $dz_t=z_{t+\Delta}-z_t$ 是前向差分,故可在 $t$ 取期望);方差等于二阶矩,$\mathbb E_t(dz_t^2)=dt$。更强的是:不仅方差等于 $dt$,而是 $dz_t^2=dt$(几乎确定地,而非仅期望意义)。直观证明:$z_{t+\Delta}-z_t\sim N(0,\Delta)$,故 $(z_{t+\Delta}-z_t)^2$ 是缩放的 $\chi^2_1$;其均值为 $\Delta$、方差为 $2\Delta^2$。前者方差 $\sim O(\Delta)$、标准差 $\sim O(\sqrt\Delta)$;但平方的方差 $\sim O(\Delta^2)$、标准差 $\sim O(\Delta)$——以 $dt$ 的阶消失。故在 $dt$ 量级上,$dz^2$ 的随机性可忽略,等于其均值 $dt$。(配套规则:$dt^2=0$,$dt\,dz=0$。)From (A.1), clearly $\mathbb E_t(dz_t)=0$ (note $dz_t=z_{t+\Delta}-z_t$ is a forward difference, so one can take the expectation at $t$); the variance equals the second moment, $\mathbb E_t(dz_t^2)=dt$. More strongly: not only does the variance equal $dt$, but $dz_t^2=dt$ (almost surely, not just in expectation). Intuitive proof: $z_{t+\Delta}-z_t\sim N(0,\Delta)$, so $(z_{t+\Delta}-z_t)^2$ is a scaled $\chi^2_1$ with mean $\Delta$ and variance $2\Delta^2$. The level has variance $\sim O(\Delta)$, standard deviation $\sim O(\sqrt\Delta)$; but the square has variance $\sim O(\Delta^2)$, standard deviation $\sim O(\Delta)$ — vanishing at order $dt$. So at the $dt$ scale, the randomness of $dz^2$ is negligible and it equals its mean $dt$. (Companion rules: $dt^2=0$, $dt\,dz=0$.)

A.2 扩散模型 / Diffusion Model

在布朗运动上加漂移 (drift) 与扩散 (diffusion) 项,构造更复杂的时序过程:

A.2 Diffusion Model

Build more complex time-series processes by adding drift and diffusion terms to the Brownian motion:

$$dx_t=\mu(\cdot)\,dt+\sigma(\cdot)\,dz_t.$$

正如离散时间叠加序列无关冲击 $\varepsilon_t$ 建出 ARMA,这里在 $dz_t$ 上叠加建出扩散模型。常见例子(各对应一个离散时间模型):

Just as discrete-time ARMA models build on serially uncorrelated shocks $\varepsilon_t$, diffusion models build on $dz_t$. Common examples (each the analogue of a discrete-time model):

$$ \begin{aligned} &\text{Random walk w/ drift:} && dx_t=\mu\,dt+\sigma\,dz_t,\\ &\text{AR(1) (Ornstein-Uhlenbeck):} && dx_t=-\phi(x_t-\mu)\,dt+\sigma\,dz_t,\\ &\text{Square-root process:} && dx_t=-\phi(x_t-\mu)\,dt+\sigma\sqrt{x_t}\,dz_t,\\ &\text{Price process:} && \frac{dp_t}{p_t}=\mu\,dt+\sigma\,dz_t. \end{aligned} $$

AR(1):漂移 $\mathbb E_t(dx_t)=-\phi(x_t-\mu)dt$ 把 $x$ 拉回稳态 $\mu$,冲击 $\sigma\,dz_t$ 推它离开。
平方根过程:波动率也随时间变 $\mathbb E_t(dx_t^2)=\sigma^2 x_t\,dt$——$x$ 趋零时波动关闭,故 $x$ 被挡在零之上(CIR 利率模型即用此)。它非线性,离散对应 $x_t=(1-\rho)\mu+\rho x_{t-1}+\sqrt{x_t}\varepsilon_t$ 非标准 ARMA、无漂亮的有限期分布——但连续时间常能给出闭式解,这正是连续时间表述的一大优势:它提供了一套有闭式解的非线性时序模型工具箱。
价格过程:让收益(价格的比例增量)序列无关,故除以 $p_t$。

一般地,$dx_t=\mu(\cdot)dt+\sigma(\cdot)dz_t$(漂移与扩散可依赖其他变量与时间)。局部均值 $\mathbb E_t(dx_t)=\mu(\cdot)dt$,局部方差 $dx_t^2=\mathbb E_t(dx_t^2)=\sigma^2(\cdot)dt$(方差等于二阶矩,因均值随 $\Delta$ 线性、其平方随 $\Delta^2$,而二阶矩随 $\Delta$)。解析方法失效时,可模拟离散化版本 $x_{t+\Delta}-x_t=\mu(\cdot)\Delta+\sigma(\cdot)\sqrt\Delta\,\varepsilon_{t+\Delta}$ 来理解。"解"一个随机微分方程,就是把它前向求解、得到未来随机变量 $x_{t+s}$ 的分布(或其条件均值、方差等特征)。

A.3 Ito 引理 / Ito's Lemma

AR(1): the drift $\mathbb E_t(dx_t)=-\phi(x_t-\mu)dt$ pulls $x$ back to its steady state $\mu$, while the shock $\sigma\,dz_t$ moves it around.
Square-root process: volatility also varies, $\mathbb E_t(dx_t^2)=\sigma^2 x_t\,dt$ — as $x$ approaches zero the volatility turns off, keeping $x$ above zero (used in the CIR interest-rate model). It is nonlinear, and its discrete analogue $x_t=(1-\rho)\mu+\rho x_{t-1}+\sqrt{x_t}\varepsilon_t$ is not standard ARMA with no pretty finite-horizon distribution — but continuous time often gives closed-form solutions, a key advantage: it provides a toolkit of nonlinear time-series models with closed-form solutions.
Price process: to make the return (proportional price increment) serially uncorrelated, divide by $p_t$.

In general, $dx_t=\mu(\cdot)dt+\sigma(\cdot)dz_t$ (drift and diffusion may depend on other variables and time). The local mean is $\mathbb E_t(dx_t)=\mu(\cdot)dt$ and the local variance $dx_t^2=\mathbb E_t(dx_t^2)=\sigma^2(\cdot)dt$ (variance equals second moment because the mean scales linearly with $\Delta$ so its square scales with $\Delta^2$, while the second moment scales with $\Delta$). When analytical methods fail, simulate the discretized version $x_{t+\Delta}-x_t=\mu(\cdot)\Delta+\sigma(\cdot)\sqrt\Delta\,\varepsilon_{t+\Delta}$. "Solving" a stochastic differential equation means solving it forward to get the distribution (or conditional mean, variance, etc.) of the future random variable $x_{t+s}$.

A.3 Ito's Lemma

Important

Ito 引理 / Ito's lemma 做二阶 Taylor 展开,只保留 $dz,dt,dz^2=dt$ 项。对 $y=f(x)$、$dx=\mu_x\,dt+\sigma_x\,dz$:$dy=f'(x)dx+\tfrac12f''(x)dx^2$。展开 $dx^2=\mu_x^2dt^2+\sigma_x^2dz^2+2\mu_x\sigma_x\,dt\,dz$,用 $dt^2=0,dz^2=dt,dt\,dz=0$ 得 $dx^2=\sigma_x^2\,dt$。故Do a second-order Taylor expansion, keeping only $dz,dt,dz^2=dt$ terms. For $y=f(x)$, $dx=\mu_x\,dt+\sigma_x\,dz$: $dy=f'(x)dx+\tfrac12f''(x)dx^2$. Expanding $dx^2=\mu_x^2dt^2+\sigma_x^2dz^2+2\mu_x\sigma_x\,dt\,dz$ and using $dt^2=0,dz^2=dt,dt\,dz=0$ gives $dx^2=\sigma_x^2\,dt$. Hence

$$dy=\left[f'(x)\,\mu_x(\cdot)+\tfrac12f''(x)\,\sigma_x^2(\cdot)\right]dt+f'(x)\,\sigma_x(\cdot)\,dz.$$

惊喜在于漂移中的第二项 $\tfrac12f''\sigma_x^2$——它捕捉的是"Jensen 不等式"效应:若 $a$ 是零均值随机变量、$b=f(a)$ 且 $f''>0$(凸),则 $b$ 的均值高于 $f(a$ 的均值$)$;$a$ 的方差越大、$f$ 越凸,$b$ 的均值越高。这一项正是连续时间定价中无处不在的 $\tfrac12\sigma^2$ 项的来源(如 Black-Scholes、对数正态期望、期限结构)。

小结 / Summary

连续时间随机微积分的全部机制可浓缩为几条直观规则:布朗运动增量 $dz\sim N(0,dt)$、独立、不可微;核心规则 $dz^2=dt$(逐路径成立,$dt^2=dt\,dz=0$);扩散过程 $dx=\mu\,dt+\sigma\,dz$ 由漂移与扩散搭出各种(含非线性平方根)模型;Ito 引理 通过保留 $dz^2=dt$ 的二阶项,给函数变换 $y=f(x)$ 的漂移添上 Jensen 项 $\tfrac12f''\sigma^2$。这套工具支撑了全书的连续时间定价。至此,Cochrane《资产定价》全书的笔记完成——从 $p=\mathbb E(mx)$ 这一个方程出发,贯通贴现因子、贝塔、均值方差三视角,经 GMM/回归/ML 的估计与检验,延伸到债券、期权、期限结构,最终落于股权溢价之谜与"真实宏观风险"这一中心问题。

The surprise is the second term in the drift $\tfrac12f''\sigma_x^2$ — it captures a "Jensen's inequality" effect: if $a$ is a mean-zero random variable and $b=f(a)$ with $f''>0$ (convex), then the mean of $b$ exceeds $f(\text{mean of }a)$; the more variance of $a$ and the more convex $f$, the higher the mean of $b$. This term is the source of the ubiquitous $\tfrac12\sigma^2$ terms in continuous-time pricing (Black-Scholes, lognormal expectations, the term structure).

Summary

The entire mechanics of continuous-time stochastic calculus condense to a few intuitive rules: Brownian-motion increments $dz\sim N(0,dt)$, independent, non-differentiable; the core rule $dz^2=dt$ (path by path, with $dt^2=dt\,dz=0$); diffusion processes $dx=\mu\,dt+\sigma\,dz$ assembling various (including nonlinear square-root) models from drift and diffusion; and Ito's lemma, which by keeping the $dz^2=dt$ second-order term adds a Jensen term $\tfrac12f''\sigma^2$ to the drift of a transformed function $y=f(x)$. This toolkit underpins the continuous-time pricing throughout the book. With this, the notes on Cochrane's Asset Pricing are complete — starting from the single equation $p=\mathbb E(mx)$, unifying the discount-factor, beta, and mean-variance views, through estimation and testing by GMM/regression/ML, extending to bonds, options, and the term structure, and ending at the equity premium puzzle and the central question of the "real macroeconomic risks" that drive prices.

习题 / Problems

若 $dp/p=\mu\,dt+\sigma\,dz$,求对数价格 $y=\ln p$ 所服从的扩散。
求 $xy$ 所服从的扩散。
设 $y=f(x,t)$,求 $y$ 的扩散表示(用 Ito 引理的多元推广)。
设 $y=f(x,w)$,$x,w$ 皆为扩散,求 $y$ 的扩散表示(记 $dz_x,dz_w$ 相关系数为 $\rho$)。
下述论证错在哪?对 $dz^2$ 用 Ito 引理得 $dz^2=2z\,dz+\tfrac12\cdot2\,dt=2z\,dz+dt$,似乎 $dz^2$ 是随机的、事后并不等于 $dt$。
模拟三条步长 $\Delta=1/10,1/100,\dots$ 的随机游走收敛到布朗运动,再画 $(z_{t+\Delta}-z_t)^2$ 的累积和,应见三例收敛到一条直线——即 $dz^2=dt$。

Problems

If $dp/p=\mu\,dt+\sigma\,dz$, find the diffusion followed by the log price $y=\ln p$.
Find the diffusion followed by $xy$.
Suppose $y=f(x,t)$. Find the diffusion representation for $y$ (the obvious multivariate extension of Ito's lemma).
Suppose $y=f(x,w)$ with both $x,w$ diffusions. Find the diffusion representation for $y$ (denote the correlation of $dz_x,dz_w$ by $\rho$).
What is wrong with the following argument? Apply Ito's lemma to $dz^2$ to get $dz^2=2z\,dz+\tfrac12\cdot2\,dt=2z\,dz+dt$, seemingly making $dz^2$ random and not equal to $dt$ after the fact.
Simulate random walks with steps $\Delta=1/10,1/100,\dots$ converging to a Brownian motion, then plot the cumulative sum of $(z_{t+\Delta}-z_t)^2$; you should see the three cases converge to a straight line — i.e. $dz^2=dt$.