初等概率论_钟开莱_英文版_ElementaryPropabilityTheory_4th.pdf

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Contents

1 Set

1.1 Sample sets

1.2 Operations with sets

1.3 Various relations

1.4 Indicator

Exercises

2 Probability

2.1 Examples of probability

2.2 Definition and illustrations

2.3 Deductions from the axioms

2.4 Independent events

2.5 Arithmetical density

Exercises

3 Counting

3.1 Fundamental rule

3.2 Diverse ways of sampling

3.3 Allocation models: binomial coefficients

3.4 How to solve it

Exercises

4 Random Variables

4.1 What is a random variable

4.2 How do random variables come about

4.3 Distribution and expectation

4.4 Integer-valued random variables

4.5 Random variables with densities

4.6 General case

Exercises

Appendix 1: Borel Fields and General Random Variables

5 Conditioning and Independence

5.1 Examples of conditioning

5.2 Basic formulas

5.3 Sequential sampling

5.4 Polya's urn scheme

5.5 Independence and relevance

5.6 Genetical models

Exercises

6 Mean, Variance, and Transforms

6.1 Basic properties of expectation

6.2 The density case

6.3 Multiplication theorem: variance and covariance

6.4 Multinomial distribution

6.5 Generating function and the like

Exercises

7 Poisson and Normal Distributions

7.1 Models for Poisson distribution

7.2 Poisson process

7.3 From binomial to normal

7.4 Normal distribution

7.5 Central limit theorem

7.6 Law of large numbers

Exercises

Appendix 2: Stirling's Formula and De Moivre-Laplace's Theorem

8 From Random Walks to Markov Chains

8.1 Problems of the wanderer or gambler

8.2 Limiting schemes

8.3 Transition probabilities

8.4 Basic structure of Markov chains

8.5 Further developments

8.6 Steady state

8.7 Winding up (or down?)

Exercises

Appendix 3 Martingale

9 Mean-Variance Pricing Model

9.1 An investments primer

9.2 Asset return and risk

9.3 Portfolio allocation

9.4 Diversification

9.5 Mean-variance optimization

9.6 Asset return distributions

9.7 Stable probability distributions

Exercises

Appendix 4 Pareto and Stable Laws

10 Option Pricing THeory

10.1 Options basics

10.2 Arbitrage-free pricing: 1-period model

10.3 Arbitrage-free pricing: N-period model

10.4 Fundamental asset pricing theorems

Exercises

General References

Answers to Problems

Values of the Standard Normal Distribution Function

Index

Doob Polya Kolmogorov Cramer

Borel Levy Keynes Feller

Contents PREFACE TO THE FOURTH EDITION PROLOGUE TO INTRODUCTION TO MATHEMATICAL FINANCE 1 SET 1.1 Sample sets 1.2 Operations with sets 1.3 Various relations 1.4 Indicator Exercises 2 PROBABILITY 2.1 Examples of probability 2.2 Deﬁnition and illustrations 2.3 Deductions from the axioms 2.4 Independent events 2.5 Arithmetical density Exercises 3 COUNTING 3.1 Fundamental rule 3.2 Diverse ways of sampling 3.3 Allocation models; binomial coeﬃcients 3.4 How to solve it Exercises xi xiii 1 1 3 7 13 17 20 20 24 31 35 39 42 46 46 49 55 62 70 vii

viii Contents 4 RANDOM VARIABLES 4.1 What is a random variable? 4.2 How do random variables come about? 4.3 Distribution and expectation 4.4 Integer-valued random variables 4.5 Random variables with densities 4.6 General case Exercises APPENDIX 1: BOREL FIELDS AND GENERAL RANDOM VARIABLES 5 CONDITIONING AND INDEPENDENCE 5.1 Examples of conditioning 5.2 Basic formulas 5.3 Sequential sampling 5.4 P´olya’s urn scheme 5.5 Independence and relevance 5.6 Genetical models Exercises 6 MEAN, VARIANCE, AND TRANSFORMS 6.1 Basic properties of expectation 6.2 The density case 6.3 Multiplication theorem; variance and covariance 6.4 Multinomial distribution 6.5 Generating function and the like Exercises 7 POISSON AND NORMAL DISTRIBUTIONS 7.1 Models for Poisson distribution 7.2 Poisson process 7.3 From binomial to normal 7.4 Normal distribution 7.5 Central limit theorem 7.6 Law of large numbers Exercises APPENDIX 2: STIRLING’S FORMULA AND DE MOIVRE–LAPLACE’S THEOREM 74 74 78 84 90 95 105 109 115 117 117 122 131 136 141 152 157 164 164 169 173 180 187 195 203 203 211 222 229 233 239 246 251

Contents ix 8 FROM RANDOM WALKS TO MARKOV CHAINS 8.1 Problems of the wanderer or gambler 8.2 Limiting schemes 8.3 Transition probabilities 8.4 Basic structure of Markov chains 8.5 Further developments 8.6 Steady state 8.7 Winding up (or down?) Exercises APPENDIX 3: MARTINGALE 9 MEAN-VARIANCE PRICING MODEL 9.1 An investments primer 9.2 Asset return and risk 9.3 Portfolio allocation 9.4 Diversiﬁcation 9.5 Mean-variance optimization 9.6 Asset return distributions 9.7 Stable probability distributions Exercises APPENDIX 4: PARETO AND STABLE LAWS 10 OPTION PRICING THEORY 10.1 Options basics 10.2 Arbitrage-free pricing: 1-period model 10.3 Arbitrage-free pricing: N-period model 10.4 Fundamental asset pricing theorems Exercises GENERAL REFERENCES ANSWERS TO PROBLEMS VALUES OF THE STANDARD NORMAL DISTRIBUTION FUNCTION INDEX 254 254 261 266 275 284 291 303 314 325 329 329 331 335 336 337 346 348 351 355 359 359 366 372 376 377 379 381 393 397

Preface to the Fourth Edition In this edition two new chapters, 9 and 10, on mathematical ﬁnance are added. They are written by Dr. Farid AitSahlia, ancien ´el`eve, who has taught such a course and worked on the research staﬀ of several industrial and ﬁnancial institutions. The new text begins with a meticulous account of the uncommon vocab- ulary and syntax of the ﬁnancial world; its manifold options and actions, with consequent expectations and variations, in the marketplace. These are then expounded in clear, precise mathematical terms and treated by the methods of probability developed in the earlier chapters. Numerous graded and motivated examples and exercises are supplied to illustrate the appli- cability of the fundamental concepts and techniques to concrete ﬁnancial problems. For the reader whose main interest is in ﬁnance, only a portion of the ﬁrst eight chapters is a “prerequisite” for the study of the last two chapters. Further speciﬁc references may be scanned from the topics listed in the Index, then pursued in more detail. I have taken this opportunity to ﬁll a gap in Section 8.1 and to expand Appendix 3 to include a useful proposition on martingale stopped at an optional time. The latter notion plays a basic role in more advanced ﬁnan- cial and other disciplines. However, the level of our compendium remains elementary, as beﬁtting the title and scheme of this textbook. We have also included some up-to-date ﬁnancial episodes to enliven, for the beginners, the stratiﬁed atmosphere of “strictly business”. We are indebted to Ruth Williams, who read a draft of the new chapters with valuable suggestions for improvement; to Bernard Bru and Marc Barbut for information on the Pareto-L´evy laws originally designed for income distributions. It is hoped that a readable summary of this renowned work may be found in the new Appendix 4. Kai Lai Chung August 3, 2002 xi

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