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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
2、试成绩：平时20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 wh
3、o have had some 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 12/19/199012/13/1996 12/13/199606/13/2001 06/13/200101/04/2006 01/04/200610/13/2008,2、独立性检验：autocorr 12/19/199012/13/1996,12/13/1996
4、06/13/2001,06/13/200101/04/2006,3、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 data r=rand(1,100); alpha=0.05; plot(r)%ru
5、ns r=r-median(r); for i=1:length(r)if r(i)=0r(i)=1;elser(i)=0;end end n1=sum(r); n2=length(r)-n1; runs=1; for i=2:length(r)if r(i)=r(i-1)runs=runs+1;end end,%testing z=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 zch=1, elseh=0, end,100 random
6、 samples of uniform distribution rand(100,1),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 price r=log(p); n=size(r,1); I=; Y=; NS=0;%caculate I N=0; for i=1:n-1if r
7、(i+1)-r(i)0I(i)=1;N=N+1;elseI(i)=0;endend,%calculate sequencesfor i=1:n-2Y(i)=I(i)*I(i+1)+(1-I(i)*(1-I(i+1);NS=NS+Y(i);end%calculate reversalsNR=n-NS;%CJ statisticsCJ=NS/NR;%calculate piepie=N/(n-1);pies=pie2+(1-pie)2;eNS=pies/(1-pies);varNS=(pies*(1-pies)+2*(pie3+(1-pie)3-pies2)/(n*(1-pies)4);delta
8、=sqrt(varNS);%final CJCJ=CJ-eNS;CJ=abs(CJ/delta);,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 winni
9、ng $100 before losing $100 in some games with 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 roun
10、d of roulette is not so foolish; there is wisdom 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
11、 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 returnin
12、g to 0 is exactly 1.,solution,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,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 .
13、 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 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=x
14、1+x2+xn tallies the gamblers winnings. Note: you 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 p
15、0pp1 and p1-p03*10-3. You should be sure to take 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
16、,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,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.,
