### Stochastic calculus [4th week, Sep]

Today when I was checking my emails, I noticed one ad from The Wharton School of the University of Pennsylvania. There was no doubt that it is the most famous business school throughout the world.  But in fact, at the first glance, I didn't realize that it was Wharton. I just thought that "oh, a school of Upeen“.

Followed its link, I was redirected to its official website, and read the introduction of its applied economics doctoral program. Although I did know that upeen would never give me an admission, I still looked carefully at the details of the program's requirements. Of course, I paid much attention on the mathematical requirements. Some familiar words came into view, like

(1) at least two courses in calculus, (2) linear algebra, (3) differential equations, and (4) probability and statistics. We also recommended that you have taken: (1) real analysis, (2) econometrics, (3) stochastic calculus.

Ok... I have one year left to meet your requirements. However, the fact is, even if I can satisfy all the basic requirements of this program, I am still not competitive enough and I have no money to pay for the tuition. Thus, the final result mentains the same: It has nothing to do with me. Moreover, I would prefer the training of theoretical economics rather than applied economics.

Here I want to say something about "stochastic". Some days ago one junior attracted my interest, not for his intelligence, but for his love of mathematics as a student of economics. I had met hundreds of schoolmates who had a good knowledge of math, either took some classes from the school of mathematics or taught themselves. But most of them were forced to enhance their math ability by the pressure from graduate schools or their advisors. Therefore, the real situation was, they did have some calculation skills after professional math trainings, but seldom did they really have the understanding of the spirit of math.

That boy was different. He didn't got high marks in every class, thereby losing the most valuable testimony of outstanding performance at study. "The only choice for me is to take part in the entrance exams for postgraduate schools ( which is called 'Kao Yan') since my lack of English and GPA ", he laughed at himself when I encouraged him to go abroad. But he told me that he had audited most of the core courses in school of math. That day we talked about a question about matrix, or more specifically, Markov chain. He offered a new solution way, which was really out of my thought. I didn't catch his meaning at first, so he explained it patiently. After that, he recommended me to listen to the course of stochastic process in school of math. To speak honestly, I had some knowledge about stochastic process so I told him. At first he was a little surprised, but soon he gave me an advice for the reason that my understanding of stochastic was not deep enough. Then he gave me a list of the courses and asked whether I would come to the class or not. Unfortunately, the time was conflicted with an other class on Tuesday afternoon. He said that it didn't matter, since I would go to another half  of the course on Thursday and the course was pretty easy. OK, I will certainly do that.

Of course, as an exchange, I provided some suggestion for the economic courses...

Now what I want to say is, why "stochastic" is so important in economics? It seems that every one has an unique answer, so do I. But today the word "stochastic calculus" really makes me confused. After searching on the Internet, I have a basic conception of it. But why it sounds so familiar? ...

Yes, Shige Peng （彭实戈）! And his backward stochastic differential equations! I see... I see...

Appendix:

Some Chinese materials about Stochastic calculus ( Author unknown, sorry! ).

1. 随机微积分（Stochastic Calculus）是干什么的？

2. 随即微积分的理论框架是怎么样建立起来的？

“收敛”，“极限”和“积分变量”都定义好了之后，我们就可以依样画葫芦，像普通微积分里面的定义那样去定义接下来的一系列概念。

3. 既然是依样画葫芦，那么跟普通微积分的区别是什么？

$int_{g=g(a)}^{g=g(b)} f(g) dg$

$int_{x=a}^{x=b} f(g(x)) g'(x) dx$

$dX/X =/= d(ln X)$

４. 随即微积分的基本运算规则是什么？

$f(W(t)) - f(W(0)) = int_0^t f'(W(u)) dW(u) + 1/2int_0^t f''(W(u)) du$

５. 关于随机微分方程

６. 随机微分方程跟偏微分方程的关系

$dX(t) = mu(t,X(t))dt + sigma(t,X(t))dW(t)$

$g(t, X(t)) = E[h(X(T))|F(t)]$.

$g_t(t,x) + mu(t,x) g_x(t, x) + 1/2 sigma^2(t,x) g_{xx}(t,x) = 0$

$g_t(t,x) + mu(t,x) g_x(t, x) + 1/2 sigma^2(t,x) g_{xx}(t,x) = rg(t,x)$

• views63 says:

说一句，网页中公式不好看。另外积分上限怎么跑到积分号前面去了；微分算子 d 、函数名没用直立体“４. 随即微……”中的公式是否有输入错？（怎么有的显示“？”）。

• cloudly says:

积分上限跑到前面是latex本来就有的一种排版模式，虽然没人觉得好看……公式可能没错，但是希腊字母没出来……我也是转载的他人文章，所以没细改。

• 谢益辉 says:

网页中公式未必不好看，但我也觉得mimeTeX的公式在很多网页中都不好看，原因是太大了，这就是为什么我自己在生成公式的代码中加入了normalsize的原因，这样公式的字号和网页字号能更匹配。

• zhangying says:

As far as I know, to get a full-covered scholarship for Master of economics in US is impossible for TOP50. It might be easier when you applies for PHD, but it's still competitive.