# 伊藤引理

## 伊藤引理较早版本

### 第一引理

${\displaystyle df(W_{t})=f'(W_{t})dW_{t}+{\frac {1}{2}}f''(W_{t})dt}$

${\displaystyle de^{W_{t}^{2}}=2W_{t}e^{W_{t}^{2}}dW_{t}+(e^{W_{t}^{2}}+2W_{t}^{2}e^{W_{t}^{2}})dt}$

### 第二引理

${\displaystyle df(t,X_{t})=\left({\frac {\partial f}{\partial t}}+{\frac {1}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}\right)d_{t}+{\frac {\partial f}{\partial x}}dX_{t}}$

### 第三引理

${\displaystyle dX_{t}=\mu _{t}\,dt+\sigma _{t}\,dW_{t}}$

${\displaystyle df(t,X_{t})=\left({\frac {\partial f}{\partial t}}+\mu _{t}{\frac {\partial f}{\partial x}}+{\frac {1}{2}}\sigma _{t}^{2}{\frac {\partial ^{2}f}{\partial x^{2}}}\right)dt+\sigma _{t}{\frac {\partial f}{\partial x}}\,dW_{t}.}$

## 到半鞅的拓展

### 连续半鞅

${\displaystyle df(X_{t})=\sum _{i=1}^{d}f_{i}(X_{t})\,dX_{t}^{i}+{\frac {1}{2}}\sum _{i,j=1}^{d}f_{i,j}(X_{t})\,d[X^{i},X^{j}]_{t}.}$

### 不连续半鞅

{\displaystyle {\begin{aligned}f(X_{t})=&f(X_{0})+\sum _{i=1}^{d}\int _{0}^{t}f_{i}(X_{s-})\,dX_{s}^{i}+{\frac {1}{2}}\sum _{i,j=1}^{d}\int _{0}^{t}f_{i,j}(X_{s-})\,d[X^{i},X^{j}]_{s}\\&{}+\sum _{s\leq t}\left(\Delta f(X_{s})-\sum _{i=1}^{d}f_{i}(X_{s-})\,\Delta X_{s}^{i}-{\frac {1}{2}}\sum _{i,j=1}^{d}f_{i,j}(X_{s-})\,\Delta X_{s}^{i}\,\Delta X_{s}^{j}\right).\end{aligned}}}

### 泊松过程

${\displaystyle dp_{s}(t)=-p_{s}(t)h(t)\,dt.}$

${\displaystyle p_{s}(t)=\exp \left(-\int _{0}^{t}h(u)\,du\right).}$

${\displaystyle d_{j}S(t)=\lim _{\Delta t\to 0}(S(t+\Delta t)-S(t^{-}))}$

${\displaystyle \eta (S(t^{-}),z)}$是跳跃幅度z概率分布，跳跃幅度的期望值是：

${\displaystyle E[d_{j}S(t)]=h(S(t^{-}))\,dt\int _{z}z\eta (S(t^{-}),z)\,dz.}$

${\displaystyle dJ_{S}(t)=d_{j}S(t)-E[d_{j}S(t)]=S(t)-S(t^{-})-(h(S(t^{-}))\int _{z}z\eta (S(t^{-}),z)\,dz)\,dt.}$

${\displaystyle d_{j}S(t)=E[d_{j}S(t)]+dJ_{S}(t)=h(S(t^{-}))(\int _{z}z\eta (S(t^{-}),z)\,dz)dt+dJ_{S}(t).}$

${\displaystyle dS(t)=\mu dt+\sigma dW(t)+d_{j}S(t).}$

{\displaystyle {\begin{aligned}g(t)-g(t^{-})&=h(t)\,dt\int _{\Delta g}\,\Delta g\eta _{g}(\cdot )\,d\Delta g+dJ_{g}(t).\end{aligned}}}

{\displaystyle {\begin{aligned}dg(t)&=\left({\frac {\partial g}{\partial t}}+\mu {\frac {\partial g}{\partial S}}+{\frac {1}{2}}\sigma ^{2}{\frac {\partial ^{2}g}{\partial S^{2}}}+h(t)\int _{\Delta g}(\Delta g\eta _{g}(\cdot )\,d{\Delta }g)\,\right)dt+{\frac {\partial g}{\partial S}}\sigma \,dW(t)+dJ_{g}(t).\end{aligned}}}

## 应用例子

### 布莱克-舒尔兹模型

${\displaystyle df=\left({\frac {\partial f}{\partial t}}+\mu S{\frac {\partial f}{\partial S}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}f}{\partial S^{2}}}\right)\,dt+\sigma S{\frac {\partial f}{\partial S}}\,dW}$

${\displaystyle df=\left({\frac {\partial f}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}f}{\partial S^{2}}}\right)\,dt+{\frac {\partial f}{\partial S}}\,dS}$

${\displaystyle df=r\left(f-S{\frac {\partial f}{\partial S}}\right)\,dt+{\frac {\partial f}{\partial S}}\,dS}$

${\displaystyle {\frac {\partial f}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}f}{\partial S^{2}}}+rS{\frac {\partial f}{\partial S}}-rf=0}$

## 參考資料

• Ito, K. (1944): Stochastic integral. Proc. Imp. Acad. Tokyo 20, 519-524.
• PROTTER, P. (1990): Stochastic Integration and Differential Equations. Springer-Verlag, Berlin.
• Black, F. & Scholes, M. (1973) :The pricing of options and corporate liabilities. J. Polit. Economy