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...


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?


在普通微积分里面,最基本的理论基础是“收敛”(convergence)和“极限”(limit)的概念,所有其他的概念都是基于这两个基本概念的。对于随机微积分,在我们建立了现代的概率论体系(基于实分析和测度论)之后,同样的我们就像当初发展普通微积分那样先建立“收敛”和“极限”这两个概念。与普通数学分析不同的是,现在我们打交道的是随机变量,比以前的普通的变量要复杂得多,相应的建立起来的“收敛”和“极限”的概念也要复杂得多。事实上,随机微积分的“收敛”不止一种,相应的“极限”也就不止一种。用的比较多的收敛概念是 convergence with probability 1 (almostsurely) 和 mean-square convergence。



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

如果g(x)是一个可导的函数,这就是我们在普通微积分中已经解决了的问题,因为dg =g'dx,所以上式可以写成:

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


比如,在普通微积分里面,有基本的微积分公式(ln x)' = 1/x

因而dx/x = d(ln x).

但在随机微积分里面则不能对dW/W 进行这样的计算

dX/X =/= d(ln X)

因为 ln(X)是不可导的。


4. 随即微积分的基本运算规则是什么?

在普通微积分里面,首先我们定义了牛顿-莱布尼兹公式f(b) - f(a) = int_a^b f'(x) dx

然后我们定义了一系列基本的运算法则,比如d(x+y) = dx + dy;d(xy) = x*dy + y*dx

和基本微积分公式,比如(x^2)' = 2x?int exp(x)dx = exp(x)。然后我们实际进行微积分运算的时候,主要是把要计算的微分或者积分按照运算法则分解成这些基本的微积分公式,然后把他们用这些基本的微积分公式套进去进行计算。



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


有了Ito公式之后,就可以计算一些基本的常用的微积分公式,比如对于f(x)=ln(x), f' = 1/x, f'' = -1/x^2s,所以ln(W(t)) - ln(W(0)) = int_0^t (1/W(u)) dW(u) + 1/2 int_0^t (-1/W(u)^2) du



5. 关于随机微分方程


比如最著名的描述股票运动的方程(其解是Geometric Brownian Motion),我们通常看到下面的形式:dS = mu S dt + sigma S dW
这个貌似微分方程的东西,其实并不是微分方程,原因很简单,S是处处不可导的,所以你不能把dt挪到左边的分母上得到一个类似于dS/dt的东西。所以这跟本就不是一个微分方程,实际上,它是如下积分方程的一个简写而已:S(t) - S(0) = int_0^t mu S dt + int_0^t sigma S dW(u)
我们通常谈论的随机“微分”方程的general的形式如下:dX(t) = mu(t,X(t))dt + sigma(t,X(t))dW(t)




随机微分方程的数值解基本上就是常/偏微分方程的数值解的拓展,比如Euler'smethod,操作起来跟常微分方程的Euler's method几乎一模一样。不同之处在于,用Euler's method解常微分方程,这种逐步往后计算每个点的值的过程只需要进行一次。而在解随机微分方程的时候,每一次只得到一个sample process,对于解一个方程,这个过程需要重复很多次。

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:


  • 谢益辉 says:


  • 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.

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