Chapter 11(stochastic process)
Chapter 2(stochastic process)
Chapter 2Conditional Expectation2.1 A Binomial Model for StockPrice Dynamicsn Figure:A three coin period binomial model.n Note that the following notation:1. Sample space2. is stock price at time k2.2 Informationn Definition 2.1 (Sets determined by the first k tosses) We say that a set A is determined by the first k coin tosses if, knowing only the outcome of the first k tosses, we can decide whether the outcome of all tosses is in A.n Note thatn1. is the collection of set determined by the first k tossesn2.n3. the random variable is -measurable, for each k=1,2,…nn Example 2.1nn Definition 2.2 (Information carried by a random variable.) Let X be a random variable We say that a set is determined by the random variable X if, knowing only the value of the random variable, we can decide whether or not . Another way of saying this is that forevery , eithern.n Note thatn 1. The collection of subsets of determined by X is a algebra, denote byn 2. If the random variable X takes finitely many different values, then is generated by thecollection of setsn these sets are called the atoms of the algebra .n 3.if X is a random variable then is given byn Example 2.22.3 Conditional Expectationn Definition 2.3 (Expectation.)n Andn We can think of as a partial average of X over the set A.n2.3.1 An examplen Let us estimate , given . Denote the estimate by .n is a random variable Y whose value at is defined byn where .n Properties of n.n.n.n.n We then take a weighted average:n Furthermore,n In conclusion, we can write n Wheren2.3.2 Definition of Conditional Expectationn Existence. There is always a random variable Y satisfying the above properties (provided thati.e., conditional expectations always exist.n Uniqueness. There can be more than one random variable Y satisfying the above properties, but if is another one, then almost surely, i.e.,n Notation 2.1 For random variables X, Y , it is standard notation to writen.n.n2.3.3 Further discussion of Partial Averagingn.n2.3.4 Properties of Conditional Expectationn We computen We can also writen A similar argument shows thatn We can verify the Tower Property,2.4 Martingalesn The ingredients are:n A super martingalen A sub martingalen A Martingalen Example 2.3 (Example from the binomial model.)n For k = 1;2 we already showed thatn The right hand side is , and so we have。
机器学习PPT(11)
Neural Networks and Learning Machines, Third Edition Simon Haykin
Copyright ©2009 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.
Figure 11.13 Clustering at various phases. The lines are equiprobability contours, p = ½ in (b), and p = ⅓ elsewhere: (a) 1 cluster (B = 0), (b) 2 clusters (B = 0.0049), (c) 3 clusters (B = 0.0056), (d) 4 clusters (B = 0.0100), (e) 5 clusters (B = 0.0156), (f) 6 clusters (B = 0.0347), and (g) 19 clusters (B = 0.0605).
Figure 11.6 Sigmoid-shaped function P(v).
Neural Networks and Learning Machines, Third Edition Simon Haykin
Copyright ©2009 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.
Copyright ©2009 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.
Chapter11布莱克休尔斯莫顿期权定价模型
衍生证券的定价。
Copyright© Zheng Zhenlong & Chen Rong, 2008
19
观察布莱克－舒尔斯微分方程，我们可以发现，受制于主观的 风险收益偏好的标的证券预期收益率并未包括在衍生证券的价值决 定公式中。这意味着，无论风险收益偏好状态如何，都不会对f的值 产生影响。因此我们可以作出一个可以大大简化我们工作的假设： 在对衍生证券定价时，所有投资者都是风险中性的。尽管这只是一 个人为的假定，但通过这种假定所获得的结论不仅适用于投资者风 险中性情况，也适用于投资者厌恶风险的所有情况。
11
由上一页的推导可知证券价格对数服从正态分布。如果
一个变量的自然对数服从正态分布，则称这个变量服从对数
正态分布。这表明ST服从对数正态分布。根据对数正态分布 的特性，以及符号的定义，我们可以得到 E(ST ) Se (T t) 和 var(ST ) S 2e 2(T t) [e 2 (T t) 1]
S S S 2 S 2 t
代入式 dG ( G a G 1 2G b 2 )dt G bdz我们就可得到 G ln S 所
x
t 2 x 2
x
遵循的随机过程为 dG d ln S ( 2 )dt dz
2
由于dlnS是股票的连续复利收益率，得出的公式说明股票的连续
复利收益率服从期望值 ( 2 )dt ，方差为 2dt 的正态分布。
2、在任意时间长度T后x值的变化也具有正态分布特征，其均值为 aT，标准差为 b T，方差为b2T。
Copyright© Zheng Zhenlong & Chen Rong, 2008
6
普通布朗运动假定漂移率和方差率为常数，若把变量 x的漂移率和方差率当作)dz
StochasticProcess
2
Discrete Time and Continuous Time Processes
•X(t) is a discrete time process if X(t) is defined only for a set of time instants tn=nT where T is a constant and n is an integer
According to Theorem 1 p( j)= [ P(0, 1)P(1,2) … P( j-1, j)]T p(0) In the Stationary case, P(0, 1)P(1,2) … P( j-1, j) = P j
Example: Random Walk With Barriers
Markov Diagram
•In the stationary form
–the Chapman-Kolmogorov equations has a graphical interpretation in terms of one-step probabilities p1,1 p1 p2 p2,2 p2,1 2 1 p1,2 p2,3 p3,2 p1,3 p3,1 3
t ,t ' pm P[ X (t ' ) n | X (t ) m] ,n
•For discrete-time process, define
i, j pm P[ X ( j ) n | X (i ) m] ,n
Markov Chain
•A stochastic process is a Markov chain if
•X(t) is a discrete value process if the set of all possible values of X(t) at all times t is a countable set SX
chapter11a
ቤተ መጻሕፍቲ ባይዱ
Inventories: Goods that have been produced, but not yet sold.
I
Changes in inventories are included as part of investment spending:
I
Assume that the amount businesses plan to spend on inventories may be di¤erent from the amount they actually spend.
The Di¤erence between Planned Investment and Actual Investment
I I
The amount of that …rms plan to spend on investment can be di¤erent from the amount they actually spend. The reason is that we need to consider inventories:
I
It is used to study the business cycle involving the interaction of many economic variables.
I
The key idea of AE model: In any particular year, the level of GDP is determined mainly by the level of AE that have several components. Economists began to study the relationship bw ‡uctuations in AE and ‡uctuations in GDP during the Great depression of the 1930s:
课程大纲-金融随机分析
附件：大纲模板研究生课程教学大纲(Course Outline)课程名称(Course Name in Chinese)：金融随机分析英文名称(Course Name in English)：Stochastic Modeling in Finance开课系财务金融系教学小组负责人马成虎开课学期□春季X 秋季学分 3一、课程的教学目的 (Course Purpose)This course is an advanced treatment of no-arbitrage approach of stochastic modeling in finance. We shall put special emphasis on continuous time modeling. Fundamental theorem and various applications in option pricing and term structure of interest rates (TSIR) will be thoroughly covered.二、教学内容及基本要求(Teaching Content and Requirements)Topics include:(a)Stochastic processes and stochastic calculus(b)Trading strategy and market span(c)No arbitrage and martingale pricing: The Fundamental Theorem(d)Black-Scholes option pricing model(e)Classical no arbitrage modeling on TSIR(f)Heath-Jarrow-Morton’s approach on TSIR(g)TSIR in presence of Levy jumps三、考核方式及要求 (Grading)There will be no final examination. Students will be assessed on the basis of class participation, a mid-term test and a term paper.Class participation 10%Mid-term test 20%Term paper 70%Total 100%四、学习本课程的前期课程要求(Required Courses in advance)Asset Pricing, Econometrics/Statistics, Optimization五、教材 (Textbook)马成虎：高级资产定价理论。
Chapter 4(stochastic process)
Chapter 4The Markov Property4.1 Binomial Model Pricing and Hedgingn Recall that is the given simple European derivative security, and the value and portfolio processes are givenby:n Example 4.1 (Lookback Option)u=2−dun The payoff is thus “path dependent ”. Working backward in time, we have:n .3)(),(;0)(),(2222==HT V HT X TH V TH Xn Example 4.2 (European Call)n consider a European call with expiration time 2 and payoff functionn Note thatn.n Define to be the value of the call at time k when Thenn.4.2 Computational Issuesn For a model with n periods, we must solve equations of the formnThere are three possible ways to deal with thisproblem:n1). Simulation.n2). Approximate a many-period model by a continuous-time model.n3). Look for Markov structure.12−n4.3 Markov Processesn Technical condition always present:We consider only functions on R and subsets of R which areBorel-measurable,n Definition 4.1 Let be a probability space. Let be a filtration under Let be a stochastic processon This process is said to be Markov if:The stochastic process is adapted to the filtration , and(The Markov Property). For each thedistribution of conditioned on is the same as the distribution of conditioned onn 4.3.1 Different ways to write the Markov propertyn(a) (Agreement of distributions). For every we haven(b) (Agreement of expectations of all functions). For every (Borel-measurable) function for which we haven Consequences of the Markov property. Let j bea positive integer.n.4.4 Showing that a process is Markovn Lemma 4.15 (Independence Lemma)n.n Example 4.5 (Showing the stock price process is Markov)n.n.4.5 Application to Exotic Optionsn Consider an n-period binomial model. Define the running maximum of the stock price to ben Consider a simple European derivative security with payoff at time n ofn Examples:n.n Lemma 5.16 The two-dimensional process is Markov.n Definen This leads us to definen Since this is a simple European option, the hedging portfolio is given by the usual formula, which in this case is。
stochastic processes 1
Springer Series in Reliability Engineering For further volumes:/series/6917Toshio NakagawaStochastic Processeswith Applications to Reliability Theory 123Dr.Toshio NakagawaDepartment of Business AdministrationAichi Institute of Technology1247Yachigusa,Yakusa-cho470-0392ToyotaJapane-mail:toshi-nakagawa@aitech.ac.jpISSN1614-7839ISBN978-0-85729-273-5e-ISBN978-0-85729-274-2DOI10.1007/978-0-85729-274-2Springer London Dordrecht Heidelberg New YorkBritish Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryÓSpringer-Verlag London Limited2011Apart from any fair dealing for the purposes of research or private study,or criticism or review,as permitted under the Copyright,Designs and Patents Act1988,this publication may only be reproduced, stored or transmitted,in any form or by any means,with the prior permission in writing of the publishers,or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency.Enquiries concerning reproduction outside those terms should be sent to the publishers.The use of registered names,trademarks,etc.,in this publication does not imply,even in the absence of a specific statement,that such names are exempt from the relevant laws and regulations and therefore free for general use.The publisher makes no representation,express or implied,with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.Cover design:eStudio Calamar,Berlin/FigueresPrinted on acid-free paperSpringer is part of Springer Science+Business Media()PrefaceI have learnedfirst reliability theory from the book Mathematical Theory of Reliability[1]written by Barlow in1965that is the most famous and excellent one theoretically up to now.I was a marvelously lucky reader,unfortunately,I could not understand throughly this book at that time because my poor mathematical tools and abysmal ignorance about stochastic processes.I have now already published three monographs Maintenance Theory of Reliability[2],Shock and Damage Models in Reliability Theory[3],and Advanced Reliability Models and Maintenance Policies[4]in which I have surveyed mainly maintenance policies in reliability theory on the research results of the author and my colleagues.Most of the three books have been written based on basic theory of stochastic processes and their mathematical tools.A number of graduate students,researchers,and engineers demand of us some book written in an easy style on stochastic processes to be able to understand readily reliability theory.Recently,plants,satellites,computer and information systems have become more large-scale and complex,and most products are distributed all over the world.If they or some of them would fail and have trouble,it might incur many serious and catastrophic situations and heavy damage.More and better mainte-nance is required constantly from the economical and environmental points as public infrastructures and operating plants become old in advanced nations. Reliability and maintenance theory is more useful for protecting such severe matters and environmental considerations,and moreover,for making good and safe living.Stochastic processes can be described by probabilistic phenomena in some space at each point of time t.The knowledge of stochastic processes and mathe-matical tools is indispensable for engineers,managers and researchers in reliability and maintenance.This book introduces basic stochastic processes from the viewpoint of elementary mathematics,mainly based on the books Stochastic Processes[5],Applied Stochastic System Modeling[6],and Introduction to Sto-chastic Processes[7].Many reliability examples are cited in this book to explain concretely stochastic processes.They are quoted mainly from the books[1–4]. Furthermore,several interesting reliability examples appeared in two famous andvvi Preface classical books[8,9].However,they have become old and their important con-tributions are going to be forgotten.This book is aimed at quoting possible examples from the two books.This book is composed of eight chapters:Chapter1is devoted to the intro-duction to stochastic processes and reliability theory.Chapters2–5are devoted to standard and fundamental stochastic processes such as Poisson processes,renewal processes,Markov chains,Markov processes,and Markov renewal processes. These four chapters are written mainly based on the books[5–7]and are intro-duced from the viewpoint of elementary mathematics without using hard proof. Chapter6is devoted to cumulative processes that are related greatly to shock and damage models in onefield of reliability.Chapter7introduces simply Brownian motion and Lévy processes,that might be omitted at thefirst reading.To under-stand and explain stochastic processes easily and actually,a lot of examples are quoted from reliability models through each chapter.Asfinal examples of reli-ability models,Chapter8takes up redundant systems and shows systematically how to use well the tools of stochastic processes to analyze them.The reader could learn both stochastic process and reliability theory from this book at the same time.I wish to thank most kindly Professor E.Çinlar,Professor S.M.Ross,and Professor S.Osaki for referring to the good books written by them and the other books in references.I wish to express my special thanks to Professor Fumio Ohi and Professor Mingchih Chen for their careful reviews of this book,and to Dr. Kodo Ito,Dr.Satoshi Mizutani,Dr.Sayori Maeji,and Mr.Xufeng Zhao for their support in writing and typing this book.Finally,I would like to express my sincere appreciation to Professor Hoang Pham,Rutgers University,and Editor Anthony Doyle,Springer-Verlag,London,for providing the opportunity for me to write this book.Toyota,April2010Toshio NakagawaReferences1.Barlow RE,Proschan F(1965)Mathematical theory of reliability.Wiley,NewYork2.Nakagawa T(2005)Maintenance theory of reliability.Springer,London3.Nakagawa T(2007)Shock and damage models in reliability theory.Springer,London4.Nakagawa T(2008)Advanced reliability models and maintenance policies.Springer,London5.Ross SM(1983)Stochastic processes.Wiley,New York6.Osaki S(1992)Applied stochastic system modeling.Springer,Berlin7.Çinlar E(1975)Introduction to stochastic processes.Prentice-Hall,Englewood Cliffs,NJ8.Takács L(1960)Stochastic processes.Wiley,New York9.Cox DR(1962)Renewal theory.Methuen,LondonContents1Introduction (1)1.1Reliability Models (2)1.1.1Redundancy (2)1.1.2Maintenance (2)1.2Stochastic Processes (2)1.3Further Studies (5)1.4Problems1 (5)References (6)2Poisson Processes (7)2.1Exponential Distribution (8)2.1.1Properties of Exponential Distribution (9)2.1.2Poisson and Gamma Distributions (14)2.2Poisson Process (18)2.3Nonhomogeneous Poisson Process (27)2.4Applications to Reliability Models (34)2.4.1Replacement at the N th Failure (34)2.4.2Software Reliability Model (36)2.5Compound Poisson Process (36)2.6Problems2 (42)References (45)3Renewal Processes (47)3.1Definition of Renewal Process (48)3.2Renewal Function (50)3.2.1Age and Residual Life Distributions (58)3.2.2Expected Number of Failures (61)3.2.3Computation of Renewal Function (64)3.3Age Replacement Policies (66)3.3.1Renewal Equation (66)viiviii Contents3.3.2Optimum Replacement Policies (69)3.4Alternating Renewal Process (73)3.4.1Ordinary Alternating Renewal Process (73)3.4.2Interval Reliability (76)3.4.3Off Distribution (79)3.4.4Terminating Renewal Process (82)3.5Modified Renewal Processes (84)3.5.1Geometric Renewal Process (84)3.5.2Discrete Renewal Process (86)3.6Problems3 (91)References (93)4Markov Chains (95)4.1Discrete-Time Markov Chain (97)4.1.1Transition Probabilities (97)4.1.2Classification of States (101)4.1.3Limiting Probabilities (103)4.1.4Absorbing Markov Chain (108)4.2Continuous-Time Markov Chain (111)4.2.1Transition Probabilities (111)4.2.2Pure Birth Process and Birth and Death Process (112)4.2.3Limiting Probabilities (119)4.3Problems4 (120)References (121)5Semi-Markov and Markov Renewal Processes (123)5.1Markov Process (124)5.2Embedded Markov Chain (132)5.2.1Transition Probabilities (133)5.2.2First-Passage Distributions (136)5.2.3Expected Numbers of Visits to States (138)5.2.4Optimization Problems (140)5.3Markov Renewal Process with Nonregeneration Points (140)5.3.1Type1Markov Renewal Process (141)5.3.2Type2Markov Renewal Process (143)5.4Problems5 (147)References (148)6Cumulative Processes (149)6.1Standard Cumulative Process (150)6.2Independent Process (155)6.3Modified Damage Models (162)6.3.1Imperfect Shock (162)6.3.2Random Failure Level (164)Contents ix6.3.3Damage with Annealing (166)6.3.4Increasing Damage with Time (169)6.4Replacement Models (170)6.4.1Three Replacement Policies (171)6.4.2Optimum Policies (173)6.5Problems6 (179)References (180)7Brownian Motion and Lévy Processes (183)7.1Brownian Motion and Wiener Processes (183)7.2Three Replacements of Cumulative Damage Models (187)7.2.1Three Damage Models (187)7.2.2Numerical Examples (189)7.3Lévy Process (191)7.4Problems7 (196)References (196)8Redundant Systems (199)8.1One-Unit System (200)8.1.1Poisson Process (200)8.1.2Nonhomogeneous Poisson Process (200)8.1.3Renewal Process (201)8.1.4Alternating Renewal Process (202)8.2Two-Unit Standby System (203)8.2.1Exponential Failure and Repair Times (204)8.2.2General Failure and Repair Times (205)8.3Standby System with Spare Units (211)8.3.1First-Passage Time to Systems Failure (211)8.3.2Expected Number of Failed Units (213)8.3.3Expected Cost and Optimization Problems (215)8.4Problems8 (216)References (217)Appendix A:Laplace Transform (219)Appendix B:Answers to Selected Problems (225)Index (249)。