## Wiener Processes and Itô's Lemma
### Itô's lemma #### [Kiyosi Itô (伊藤清)](https://en.wikipedia.org/wiki/Kiyosi_It\%C3\%B4) <img align="center" style="padding-right:10px;" width=70% src="../myfig/options/ito.jpg"> ---
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$$\Delta G=\textcolor{blue}{\frac{\partial G}{\partial x}\Delta x+\frac{\partial G}{\partial t}\Delta t}+\textcolor{green}{\frac{1}{2}\frac{\partial^2 G}{\partial x}(\Delta x)^2}+\textcolor{red}{\frac{\partial^2 G}{\partial x\partial t}\Delta x\Delta t+\frac{1}{2}\frac{\partial^2 G}{\partial t}(\Delta t)^2}+o(\Delta t)$$ $$\textcolor{red}{\Delta x\Delta t}=(a\Delta t+b\Delta z)\Delta t=a\textcolor{red}{(\Delta t)^2}+b\epsilon\textcolor{red}{\Delta t\sqrt{\Delta t}}=o(\Delta t)$$ $$\textcolor{red}{(\Delta t)^2}=o(\Delta t)$$ $$\textcolor{green}{(\Delta x)^2}=a^2\textcolor{red}{(\Delta t)^2}+2ab\Delta z\Delta t+b^2(\Delta z)^2=a^2\textcolor{red}{(\Delta t)^2}+2ab\epsilon\textcolor{red}{\Delta t\sqrt{\Delta t}}+\textcolor{green}{b^2\epsilon^2\Delta t}$$ $$\mathbb{E}[\epsilon^2\Delta t]=(Var[\epsilon]+(\mathbb{E}[\epsilon])^2)\Delta t=\Delta t$$ $$Var[\epsilon^2\Delta t]=（\mathbb{E}[\epsilon^4]-(\mathbb{E}[\epsilon^2])^2）\Delta t=（(4-1)\mathbb{E}[\epsilon^{4-2}]-1）\Delta t=2\Delta t$$
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$$\int_{-\infty}^\infty t^4\frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}dt$$
--- ### The Stock Price Assumption - Consider a stock whose price is $S$ - In a short period of time of length $\Delta t$, the return on the stock is normally distributed: $$\frac{\Delta S}{S}\approx\phi(\mu\Delta t,\sigma^2\Delta t)$$ - So the price $S_T$ is lognormal distributed $$\ln S_T-\ln S_0\approx\phi\left[\left(\mu-\frac{\sigma^2}{2}\right)T,\sigma^2T\right]$$ or $$\ln S_T\approx\phi\left[\ln S_0+\left(\mu-\frac{\sigma^2}{2}\right)T,\sigma^2T\right]$$