# 伊藤引理

## 伊藤引理较早版本

### 第一引理

$df(W_t) = f'(W_t)dW_t + \frac{1}{2} f''(W_t)dt$

$de^{W_t^2}= 2W_t e^{W_t^2} dW_t + (e^{W_t^2} + 2W_t^2 e^{W_t^2} )dt$

### 第二引理

$df = \frac{\partial f}{\partial W_t} dW_t + \left(\frac{\partial f}{\partial t} + \frac{1}{2} \frac{\partial^2 f}{\partial W_t^2}\right)dt$

### 第三引理

$dX_t = f(X_t , t) dW_t + g(X_t, t) dt$

$dh = \frac{\partial h}{\partial X_t}f(X_t ,t) dW_t + \left\{\frac{\partial h}{\partial t} + \frac{\partial h}{\partial X_t}g(X,t) + \frac{1}{2}\frac{\partial^2 h}{\partial X_t^2}(f(X_t ,t))^2 \right\} dt$}}

## 到半鞅的拓展

### 连续半鞅

$df(X_t) = \sum_{i=1}^d f_{i}(X_t)\,dX^i_t + \frac{1}{2}\sum_{i,j=1}^df_{i,j}(X_t)\,d[X^i,X^j]_t.$

### 不连续半鞅

\begin{align} f(X_t)= & f(X_0)+\sum_{i=1}^d\int_0^t f_{i}(X_{s-})\,dX^i_s + \frac{1}{2}\sum_{i,j=1}^d \int_0^t f_{i,j}(X_{s-})\,d[X^i,X^j]_s\\ &{} + \sum_{s\le t}\left(\Delta f(X_s)-\sum_{i=1}^df_{i}(X_{s-})\,\Delta X^i_s-\frac{1}{2}\sum_{i,j=1}^d f_{i,j}(X_{s-})\,\Delta X^i_s \, \Delta X^j_s\right). \end{align}

### 泊松过程

$d p_s(t) = -p_s(t) h(t) \, dt.$

$p_s(t) = \exp \left(-\int_0^t h(u) \, du \right).$

$d_j S(t)=\lim_{\Delta t \to 0}(S(t+\Delta t)-S(t^-))$

$\eta(S(t^-),z)$是跳跃幅度z概率分布，跳跃幅度的期望值是：

$E[d_j S(t)]=h(S(t^-)) \, dt \int_z z \eta(S(t^-),z) \, dz.$

$d J_S(t)=d_j S(t)-E[d_j S(t)]=S(t)-S(t^-)-(h(S(t^-)) \int_z z \eta(S(t^-),z) \, dz) \, dt.$

$d_j S(t) = E[d_j S(t)] + d J_S(t) = h(S(t^-)) (\int_z z \eta(S(t^-),z) \, dz) dt + d J_S(t).$

$d S(t) = \mu dt + \sigma dW(t) + d_j S(t).$

\begin{align} g(t)-g(t^-) & =h(t) \, dt \int_{\Delta g} \, \Delta g \eta_g(\cdot) \, d\Delta g + d J_g(t). \end{align}

\begin{align} d g(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 \, d W(t) + d J_g(t). \end{align}

## 应用例子

$df(t,S_t) = \left(\frac{\partial f}{\partial t} + \frac{1}{2}\left(S_t\sigma\right)^2\frac{\partial^2 f}{\partial S^2}\right)\,dt +\frac{\partial f}{\partial S}\,dS_t.$
$dV_t = r\left(V_t-\frac{\partial f}{\partial S}S_t\right)\,dt + \frac{\partial f}{\partial S}\,dS_t.$
$\frac{\partial f}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 f}{\partial S^2} + rS\frac{\partial f}{\partial S}-rf = 0.$

## 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