1、随机过程,上海财经大学金融学院韩其恒,参考文献,J. Michael Steele (2003): Stochastic Calculus and Financial Application. Springer张波（2000）：应用随机过程。中国人民大学出版社邵芋（2003）：微观金融学及其数学基础（第二部分）。清华大学出版社Ioannis Karatzas, Steven E. Shreve (1987), Brownian Motion and Stochastic Calculus. Springer刘嘉昆（2000）：应用随机过程。科学出版社,教案网址,教材pdf，课件考试成绩：平时2
2、0分；课堂报告：20分；期末考试60分,基础知识,微积分概率论数理统计实变函数泛函分析英语,前言,This book is designed for students who want to develop professional skill in stochastic calculus and its application to problems in finance. The Wharton School course that forms the basis for this book is designed for energetic students who have had s
3、ome experience with probability and statistics.,内容,随机游走与一步分析法鞅初步布郎运动鞅路径伊藤积分局部鞅与伊藤积分伊藤公式随机微分方程套利与随机微分方程,第一章随机游走与第一步分析法,随机游走的检验,1、回归检验,对上证综合指数的回归检验12/19/199012/13/199612/13/199606/13/200106/13/200101/04/200601/04/200610/13/2008,2、独立性检验：autocorr12/19/199012/13/1996,12/13/199606/13/2001,06/13/200101/04/
4、2006,3、游程检验,检验时间序列是否独立。,（SHCI的0/1转换表，90-01）,游程检验,列出观测值计算观测值中值将观测值中大于中值的记为+，小于中值的记为-，等于中值的记为+。正号的个数记为n1，负号的个数记为n2。按照观测值的顺序，将正负号的改变个数加1记为RANS。,当n1或n2大于20时，以下统计量近似服从标准正态分布。,当Z0值小于1.96时，在0.05置信水平上不能拒绝观测值是独立的这一原假设。,clear%importing datar=rand(1,100);alpha=0.05;plot(r)%runsr=r-median(r);for i=1:length(r) i
5、f r(i)=0 r(i)=1; else r(i)=0; endendn1=sum(r);n2=length(r)-n1;runs=1;for i=2:length(r) if r(i)=r(i-1) runs=runs+1; endend,%testingz=abs(runs-2*n1*n2/(n1+n2)+1)/sqrt(2*n1*n2*(2*n1*n2-n1-n2)/(n1+n2)2*(n1+n2-1);c=norminv(1-alpha/2);if zc h=1,else h=0,end,100 random samples of uniform distributionrand(1
6、00,1),Z=0.4020,1000 random samples from AR(1) process rt+1= rt+t, =0.5, r1=0.5,Runs test: z=11.4532,SHCI,4、顺逆检验(Sequences and reversals),Cowles和Jones，1937年，CJ统计量,function CJ=funCJ(p) %log pricer=log(p);n=size(r,1);I=;Y=;NS=0;%caculate IN=0;for i=1:n-1 if r(i+1)-r(i)0 I(i)=1; N=N+1; else I(i)=0; end
7、end,SHCI,5、鞅过程的检验：广义谱分析,SHCI（周）,第一章 随机游走与一步分析法,随机游走,分布,财富,财富第一次达到A或-B的时间,问题,约束条件：,结论：,第一节 第一步分析法,第二节 时间与无限,矩的有限性,1(A) 表示事件A的指示函数，Z是一个非负且取值为正整数的随机变量，则,的均值：一步分析法,边界条件是,边界条件是,example,第三节 非公平游戏,有偏随机游走,的均值,边界条件是,第四节 数值计算与直觉,We compute the probability of winning $100 before losing $100 in some games with
8、odds that are typical of the worlds casinos. The table assumes a constant bet size of $1 on all rounds of the game.,One of the lessons we can extract from this table is that the traditional movie character who chooses to wager everything on a single round of roulette is not so foolish; there is wisd
9、om to back up the bravado.,第五节 停时的分布,概率生成函数（probability generating function）,性质：独立随机变量和的概率生成函数是随机变量概率生成函数的积。,第一步分析法,Exercises,Consider simple random walk beginning at 0 and show that for any L0 the expected number of visits to level L before returning to 0 is exactly 1.,鞅理论指出在公平游戏中，无论一个赌博者多么聪明，都不可能获
10、得超额盈利。,第二章 鞅,定义：一个随机变量序列Mn：0=n是关于随机变量Xn：0=n=0，E(Xn)=1，则,例4,如果Yn是独立同分布的随机变量，其矩生成函数(矩母函数,moment generating function)满足,缩写,Exercises,Consider simple random walk beginning at 0 and show that for any k0 the expected number of visits to level k before returning to 0 is exactly 1.,solution,第二节 New Martinga
11、le from old,鞅转换定理,停时,在公平游戏中,赌博者决定何时停止赌博只能以他已经赌过的结果为依据,而不能以以后的结果决定停时。,停时过程,Exercise 2.1 (Finding a martingale),Consider a gambling game with multiple payouts: the player loses $1 with probability , wins $1 with probability , and wins $2 with probability . Specifically, we assume that =0.52, =0.45, =0
12、.03, so the expected value of each round of the game is only $-0.01.(a)Suppose the gambler bets one dollar on each round of the game and that xk is the amount won or lost on the kth round. Find a real number x such that Mn=xSn is a martingale where Sn=x1+x2+xn tallies the gamblers winnings. Note: yo
13、u will need to find the numerical solutions to a cubic equation, but x=1 is one solution so the cubic can be reduced to a quadratic.(b)Let p denotes the probability of winning $100 or more before losing $100. Give numerical bounds p0 and p1 such that p0pp1 and p1-p03*10-3. You should be sure to take
14、 proper account of the fact that the gamblers fortune may skip over $100 if a win of $2 takes place when the gamblers fortune is $99. This issue of “overshoot” is one that comes up in many problems where a process can skip over intervening states.,停时定理,第三节 再论一步分析法,随机游走,分布,财富,财富第一次达到A或-B的时间,结论,一致收敛定理
15、,重新证明,由于Sn是一个鞅,根据停时定理,Sn也是一个鞅,停时的均值,随机游走,分布,财富,财富第一次达到A或-B的时间,结论,重新证明,有偏随机游走,结论,重新证明,第四节 下鞅,凸函数的性质,Jensen 不等式,例,Lp空间,Lp空间与Jensen 不等式,第五节 多布(Doob)不等式,指标移动(Index Shifting)不等式,Observation: many random variables are best understood when written in terms of the values of an associated stopping time.,最大序列
16、的尾部概率,多布最大不等式(Doobs Maximal Inequality),最大序列的尾部概率,指标移动(Index Shifting)不等式,自改进不等式(Self-improving Inequality),Hlder不等式,多布Lp不等式,由多布最大不等式知,第六节 鞅收敛,多布最大不等式,局部化的思想,Fatou(法图)引理,极限的有界性,Sticking with a Good Game,上交(up-crossings),上交不等式(up-crossing Inequality),L1有界鞅,收敛定理的一个有趣结果,练习,第三章 布郎运动,BachelierEinsteinAs
17、a creation of pure mathematics，it is an entity of uncommon beauty. It reflects a perfection that seems closer to a law of nature than to a human invention.,标准布郎运动的定义,第一节 协方差与特征函数,多元高斯随机变量,多元高斯分布的定义,独立性,特征函数,如果X是任意一个随机变量,如果V是任意一个d-维随机向量,注：密度函数与特征函数相互唯一确定。如果X是一个均值为，方差是2的高斯变量,如果V是一个均值为，协方差矩阵是的高斯向量,协方差函
18、数与高斯过程,布朗运动的协方差函数,高斯过程的均值函数与协方差函数,利用密度函数与特征函数的相互唯一确定性，可知Z服从一元高斯分布。,第二节 序列逼近,L20,1上的标准完备正交系,Parseval恒等式,Fourier transformation,x=sin(2*pi*0:n/10);,x=sin(2*pi*1:n/12)+2*sin(2*pi*1:n/20+pi/3);,x=sin(2*pi*1:n/12)+2*sin(2*pi*1:n/20+pi/3)+randn(1,n),x=sin(2*pi*1:n/12)+2*sin(2*pi*1:n/20+pi/3)+3*randn(1,n),
19、fft; n=600; x=randn(1,n);,稳定分布简介,韩其恒上海财经大学金融学院2003年12月15日,稳定分布的定义,称随机变量X服从稳定分布，如果对于任意的正数A与B，存在着一个正数C与一个实数D，使得,中心极限定理,稳定分布的特征函数,记为,随机过程复习,第一章 随机游走与第一步分析法,随机游走,分布,财富,财富第一次达到A或-B的时间,问题,第一节 第一步分析法,约束条件：,结论：,的均值：一步分析法,边界条件是,边界条件是,第三节 非公平游戏,有偏随机游走,概率生成函数,性质：独立随机变量和的概率生成函数是随机变量概率生成函数的积。,probability generat
20、ing function,鞅理论指出在公平游戏中，无论一个赌博者多么聪明，都不可能获得超额盈利。,第二章 鞅,定义：一个随机变量序列Mn：0=n是关于随机变量Xn：0=n=0，E(Xn)=1，则,矩母函数,moment generating function,第二节 New Martingale from old,鞅转换定理,停时,在公平游戏中,赌博者决定何时停止赌博只能以他已经赌过的结果为依据,而不能以以后的结果决定停时。,停时过程,停时定理,一致收敛定理,Exercise 2.1 (Finding a martingale),Consider a gambling game with mu
21、ltiple payouts: the player loses $1 with probability , wins $1 with probability , and wins $2 with probability . Specifically, we assume that =0.52, =0.45, =0.03, so the expected value of each round of the game is only $-0.01.(a) Suppose the gambler bets one dollar on each round of the game and that
22、 xk is the amount won or lost on the kth round. Find a real number x such that Mn=xSn is a martingale where Sn=x1+x2+xn tallies the gamblers winnings. (b) Let p denotes the probability of winning $100 or more before losing $100. Give numerical bounds p0 and p1 such that p0pp1 and p1-p03*10-3.,(a) 解：
23、由鞅的定义,(b) 解：,第四节 下鞅,Jensen 不等式,例,Lp空间,第五节 多布(Doob)不等式,多布最大不等式(Doobs Maximal Inequality),第六节 鞅收敛,Exercise 3.2Cautionary Tale: Covariance and Independence,The notions of independence and covariance are less closely related than elementary courses sometimes lead one to suspect. Give an example of rando
24、m variable X and Y with Cov(X, Y)=0 such that both X and Y are Gaussian yet X and Y are not independent. Naturally, X and Y cannot be jointly Gaussian.,例子,随机变量Z的密度函数为：P(Z=1)=0.5; P(Z=-1)=0.5，随机变量Y是服从均值为0，方差为1的正态分布，Z与Y相互独立。令X=ZY，则随机变量X与Y服从正态分布，并且Cov(X,Y)=0，但是X与Y却不服从联合正态分布。,第三节 两个小波,母波,分形的美,母波,Hn(t)是L
25、20,1上的标准完备正交系,clearn=500;t=0:1/n:1;%input j and kj=0;for k=0:2j-1 for i=1:n+1 x=2j*t(i)-k; if x=0 end end subplot(2j,1,k+1) plot(t,h,r.) hold on plot(0,1,0,0) hold offend,母波,Hn(t)是L20,1上的标准完备正交系,两个小波之间的关系,第四节 布郎运动的小波表达式,布郎运动的模拟,clear%partition 0,1 num=500;t=0:1/num:1;%exponent number of summationJ=1
26、5;%random standard normal distributionz=randn(1,2J);%summation of n=0x=z(1)*t;z=z(2:end);,%summation of n from 1 to 2J-1for j=0:J-1 for k=0:2j-1 %compute delta for i=1:num+1 flag=2j*t(i)-k; if flag=0 endend%plotplot(t,x),num=500; n=23-1,num=1000; n=23-1,num=500; n=25-1,num=1000; n=25-1,num=500; n=27
27、-1,num=1000; n=27-1,num=500; n=210-1,num=1000; n=210-1,num=500; n=215-1,num=1000; n=215-1,随机过程复习,第三章 布郎运动,标准布郎运动的定义,多元高斯分布的定义,独立性,特征函数,如果X是任意一个随机变量,如果V是任意一个d-维随机向量,注：密度函数与特征函数相互唯一确定。如果X是一个均值为，方差是2的高斯变量,如果V是一个均值为，协方差矩阵是的高斯向量,布朗运动的协方差函数,第二节 序列逼近,L20,1上的标准完备正交系,Parseval恒等式,Hn(t)是L20,1上的标准完备正交系,第三节 两个小波,第四节 布郎运动的小波表达式,
