37. Ito's Lemma
Itô 引理 (Ito's lemma) 是随机微积分的链式法则。对 \(f\in C^2\) 与 SDE \(dX_t=m_t\,dt+\sigma_t\,dZ_t\)，有 (37.1) \(df(X_t)=f'(X_t)(m_t\,dt+\sigma_t\,dZ_t)+\frac12f''(X_t)\sigma_t^2\,dt\)——比普通链式法则多出一个 \(\frac12f''\sigma_t^2\,dt\) 的二阶项，源于二阶 Taylor 展开 (37.2) 加上微积分法则 Lemma 37.1：\((dt)^p=0\ (p>1)\)、\((dZ_t)^2=dt\)、\(dZ_t\,dt=0\)。关键在 \((dZ_t)^2=dt\)（因 \(dZ_t\sim\mathcal N(0,dt)\)，\((dZ_t)^2/dt\sim\chi_1^2\)，均值 \(1\)）。应用：几何布朗运动 (GBM) \(dX_t=\mu X_t\,dt+\sigma X_t\,dZ_t\) 取对数得 \(d\log X_t=(\mu-\frac12\sigma^2)dt+\sigma\,dZ_t\)（带漂移的布朗运动，故 GBM 恒正，Remark 37.1）。多元情形 (37.2 多元版) 多出交叉项 \(f_{XY}(dX_t)(dY_t)\)；二维 GBM 用 \(dZ_t^S dZ_t^C=\rho\,dt\) 求 \(d(S_tC_t)\) (37.4)。
Ito's lemma is the chain rule of stochastic calculus. For \(f\in C^2\) and the SDE \(dX_t=m_t\,dt+\sigma_t\,dZ_t\), we have (37.1) \(df(X_t)=f'(X_t)(m_t\,dt+\sigma_t\,dZ_t)+\frac12f''(X_t)\sigma_t^2\,dt\) — one extra second-order term \(\frac12f''\sigma_t^2\,dt\) relative to the ordinary chain rule, arising from the second-order Taylor expansion (37.2) plus the calculus rules in Lemma 37.1: \((dt)^p=0\ (p>1)\), \((dZ_t)^2=dt\), \(dZ_t\,dt=0\). The key is \((dZ_t)^2=dt\) (since \(dZ_t\sim\mathcal N(0,dt)\), \((dZ_t)^2/dt\sim\chi_1^2\) with mean \(1\)). Application: geometric Brownian motion (GBM) \(dX_t=\mu X_t\,dt+\sigma X_t\,dZ_t\) taking logs gives \(d\log X_t=(\mu-\frac12\sigma^2)dt+\sigma\,dZ_t\) (a Brownian motion with drift, so GBM stays positive, Remark 37.1). The multivariate case (multivariate version of 37.2) adds a cross term \(f_{XY}(dX_t)(dY_t)\); a 2D GBM uses \(dZ_t^S dZ_t^C=\rho\,dt\) to solve \(d(S_tC_t)\) (37.4).
37.1 Univariate Case of Ito's Lemma
Theorem 37.1（Itô 引理）：设 \(f:\mathbb R\to\mathbb R\) 且 \(f\in C^2\)（二阶导数连续）。若随机过程 \(\{X_t\}\) 满足 SDE \(dX_t=m_t\,dt+\sigma_t\,dZ_t\)（\(X_0\) 给定），则有 (37.1)：
Theorem 37.1 (Ito's lemma): let \(f:\mathbb R\to\mathbb R\) with \(f\in C^2\) (second-order derivative continuous). If a stochastic process \(\{X_t\}\) satisfies the SDE \(dX_t=m_t\,dt+\sigma_t\,dZ_t\) (\(X_0\) given), then (37.1):
$$df(X_t)=f'(X_t)(m_t\,dt+\sigma_t\,dZ_t)+\tfrac12f''(X_t)\sigma_t^2\,dt\tag{37.1}$$
证明的出发点是二阶 Taylor 展开 (37.2)：\(df(X_t)=f'(X_t)\,dX_t+\frac12f''(X_t)(dX_t)^2\)，再用下面的微积分法则。
Lemma 37.1（微积分法则） 以下法则都以 \(dt\) 为基准比较： (1) \((dt)^p=0\) 对 \(\forall p>1\)；(2) \((dZ_t)^2=dt\)；(3) \(dZ_t\,dt=0\)。
以下证明是启发式的（直觉用，非严格；严格处理见 He 2019d）： - (1)：\((dt)^p=(dt)^{p-1}dt=dt\cdot\lim_{N\to\infty}(T/N)^{p-1}=dt\cdot0=0\)。 - (2)：\(dZ_t\sim\mathcal N(0,dt)\) 蕴含 \((dZ_t)^2/dt\sim\chi_1^2\)，由卡方分布性质 \(\mathbb E[(dZ_t)^2/dt]=1\Rightarrow\mathbb E[(dZ_t)^2]=dt\)，\(\operatorname{Var}((dZ_t)^2/dt)=2\Rightarrow\operatorname{Var}((dZ_t)^2)=2(dt)^2=0\)，故 \((dZ_t)^2=dt\)（确定量）。 - (3)：由 (2)，\(dZ_t\approx(dt)^{1/2}\)，故 \(dZ_t\,dt\approx(dt)^{3/2}\)，由 (1) 为 \(0\)。
把 (37.2) 展开并代入 Lemma 37.1：
The proof starts from the second-order Taylor expansion (37.2): \(df(X_t)=f'(X_t)\,dX_t+\frac12f''(X_t)(dX_t)^2\), then uses the calculus rules below.
Lemma 37.1 (calculus rules) All rules are compared against \(dt\): (1) \((dt)^p=0\) for \(\forall p>1\); (2) \((dZ_t)^2=dt\); (3) \(dZ_t\,dt=0\).
The following proof is heuristic (for intuition, not rigorous; rigorous treatment in He 2019d): - (1): \((dt)^p=(dt)^{p-1}dt=dt\cdot\lim_{N\to\infty}(T/N)^{p-1}=dt\cdot0=0\). - (2): \(dZ_t\sim\mathcal N(0,dt)\) implies \((dZ_t)^2/dt\sim\chi_1^2\); by properties of the chi-square distribution \(\mathbb E[(dZ_t)^2/dt]=1\Rightarrow\mathbb E[(dZ_t)^2]=dt\), and \(\operatorname{Var}((dZ_t)^2/dt)=2\Rightarrow\operatorname{Var}((dZ_t)^2)=2(dt)^2=0\), so \((dZ_t)^2=dt\) (a deterministic quantity). - (3): by (2), \(dZ_t\approx(dt)^{1/2}\), so \(dZ_t\,dt\approx(dt)^{3/2}\), which is \(0\) by (1).
Expanding (37.2) and substituting Lemma 37.1:
$$df(X_t)=f'(X_t)(m_t\,dt+\sigma_t\,dZ_t)+\tfrac12f''(X_t)\Big[\underbrace{m_t^2(dt)^2}_{=0}+\underbrace{2m_t\sigma_t\,dZ_t\,dt}_{=0}+\underbrace{\sigma_t^2(dZ_t)^2}_{=\sigma_t^2dt}\Big]\tag{37.2}$$
37.2 Example 37.1 — Geometric Brownian Motion
给定 \(dX_t=\mu X_t\,dt+\sigma X_t\,dZ_t\)，求 \(d\log X_t\)（\(X_0>0\)）。记 \(f(x)=\log x\)，则 \(f'(x)=1/x\)、\(f''(x)=-1/x^2\)。代入 Itô 引理 (37.1)（漂移 \(m_t=\mu X_t\)、波动 \(\sigma_t=\sigma X_t\)）：
Given \(dX_t=\mu X_t\,dt+\sigma X_t\,dZ_t\), solve for \(d\log X_t\) (\(X_0>0\)). Denote \(f(x)=\log x\), then \(f'(x)=1/x\), \(f''(x)=-1/x^2\). Substituting into Ito's lemma (37.1) (with drift \(m_t=\mu X_t\), volatility \(\sigma_t=\sigma X_t\)):
$$df(X_t)=\frac1{X_t}(\mu X_t\,dt+\sigma X_t\,dZ_t)-\frac12\frac1{X_t^2}\sigma^2X_t^2\,dt=\Big(\mu-\frac12\sigma^2\Big)dt+\sigma\,dZ_t$$
这是一个带漂移的布朗运动，故 \(\{X_t\}\) 称为几何布朗运动 (geometric Brownian motion, GBM)。
Remark 37.1 对 \(X_t\) 取对数时隐含假设它恒正。这是对的：因 \(X_0>0\)，过程从零以上出发；若它将跌破零，则必有一个先到的瞬间它恰为零（路径连续）。记该首达零时刻为 \(t_0\)，则由 GBM 的形式 \(dX_t=\mu X_t\,dt+\sigma X_t\,dZ_t\)，在 \(X_{t_0}=0\) 处 \(dX_t=0\) 对 \(\forall t\ge t_0\)，于是它将永远停在零，绝不为负。
37.3 Multivariate Case of Ito's Lemma
多元情形的二阶 Taylor 展开为：
This is a typical Brownian motion with drift, so \(\{X_t\}\) is called geometric Brownian motion (GBM).
Remark 37.1 Taking the log of \(X_t\) implicitly assumes it is always positive. This is true: since \(X_0>0\), the process starts above zero; if it were ever to go below zero, there must be one point at which it is exactly zero before going negative (continuous paths). Denote that first zero-moment \(t_0\); then by the form of GBM \(dX_t=\mu X_t\,dt+\sigma X_t\,dZ_t\), at \(X_{t_0}=0\) we have \(dX_t=0\) for all \(t\ge t_0\), so it stays at zero forever and can never be negative.
37.3 Multivariate Case of Ito's Lemma
In the multivariate case the second-order Taylor expansion is:
$$df(X_t,Y_t)=f_X\,dX_t+f_Y\,dY_t+\tfrac12f_{XX}(dX_t)^2+\tfrac12f_{YY}(dY_t)^2+f_{XY}(dX_t)(dY_t)$$
只需正确定义 \(f(x,y)\) 并代入 SDE，借助 Lemma 37.1 的法则即得多元 Itô 引理。
Example 37.2（二维几何布朗运动）：给定 \(dS_t=\mu_S S_t\,dt+\sigma_S S_t\,dZ_t^S\)、\(dC_t=\mu_C C_t\,dt+\sigma_C C_t\,dZ_t^C\)，且 \(dZ_t^S dZ_t^C=\rho\,dt\)（相关系数 \(\rho\)）。求 \(d(S_tC_t)\)。取 \(f(x,y)=xy\)，则 \(f_x=y\)、\(f_y=x\)、\(f_{xx}=f_{yy}=0\)、\(f_{xy}=1\)。代入并用 Lemma 37.1 得 (37.4)：
One just needs to properly define \(f(x,y)\) and plug in the SDE; with the rules in Lemma 37.1 we reach the multivariate Ito's lemma.
Example 37.2 (Two-dimensional GBM): given \(dS_t=\mu_S S_t\,dt+\sigma_S S_t\,dZ_t^S\), \(dC_t=\mu_C C_t\,dt+\sigma_C C_t\,dZ_t^C\), with \(dZ_t^S dZ_t^C=\rho\,dt\) (correlation \(\rho\)). Solve for \(d(S_tC_t)\). Take \(f(x,y)=xy\), then \(f_x=y\), \(f_y=x\), \(f_{xx}=f_{yy}=0\), \(f_{xy}=1\). Substituting and using Lemma 37.1 gives (37.4):
$$d(S_tC_t)=C_tS_t(\mu_S\,dt+\sigma_S\,dZ_t^S)+C_tS_t(\mu_C\,dt+\sigma_C\,dZ_t^C)+\sigma_S\sigma_C S_tC_t\,\rho\,dt\tag{37.4}$$
References
- He, X. (2019d). Stochastic Calculus Notes by Xindi He.
- He, X. (2020–2024). Asset Pricing (lecture notes), Ch. 37.