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《初等概率论》是一部介绍概率论及其应用的入门教程。其原始版本面世已经有30余年，但仍然是本科一二年级的经典概率教程。在第4版中增加了两章讲述应用和数学金融，传承之前版本详细、严谨的风格，讲述了有价证券和期货理论的基本知识。书中用最初等的方法讲述了概率测度、随机变量、分布以及期望等基本概念。离散和连续的案例都有所涉及，在讲述后者的时候运用了微积分知识。配以大量的典型例子重点讲述概率推理，集中介绍了组合问题、Poison过程、随机漫步、遗传模型和Markov链。每章末都附有习题及其解答。

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

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 P61yas 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: STIRLINGS FORMULA AND DE MOIVRE-LAPLACES 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

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In probability theory the notion of a set plays a more fundamental

role. Furthermore we are interested in very general kinds of sets as well as

specific concrete ones. To begin with the latter kind, consider the following

examples:

(a) a bushel of apples;

(b) fifty-five cancer patients under a certain medical treatment;

(c) all the students in a college;

(d) all the oxygen molecules in a given container;

(e) all possible outcomes when six dice are rolled;

(f) all points on a target board.

Let us consider at the same time the following “smaller” sets:

(a') the rotten apples in that bushel;

(b') those patients who respond positively to the treatment;

(c') the mathematics majors of that college;

(d') those molecules that are traveling upwards;

(e') those cases when the six dice sh...

At their most fundamental level the financial instruments we deal with here are of either equity-type or debt-type. Equity-type instruments represent ownership of companies. In this chapter, and for simplicity, we will consider publicly traded stocks, on stock markets such as the NYSE, as the only equity-type securities. Many of the very large and well-established companies pay dividends to their stock holders. These are payments made, generally quarterly, sometimes less frequently, by the companies. Debt-type instruments represent cash loans. Investors lend money to corporations or to the government (both federal and local) by purchasing bonds. As with any loan, an interest payment schedule (so-called coupon payment) is specified as well as a date (so-called maturity date) by which the lo...

An option is a contract giving its holder the right, without the obligation, to either buy or sell a security such as a stock (the underlying security), at a predetermined price (the exercise or strike price). If the right granted is to buy, then the option is labeled a call. If the right granted is to sell, the option is labeled a put. This type of contract comes with an expiration or maturity date, which may be infinite. Options that are exercisable only at expiration are called European, and those that can be exercised any time up to expiration are called American. Both American and European option types are traded worldwide, but American options are more common. Options are the most representative type of contingent claims.

Option contracts can be used for several purposes. Prices of...

书籍介绍

《初等概率论》是一部介绍概率论及其应用的入门教程。其原始版本面世已经有30余年，但仍然是本科一二年级的经典概率教程。在第4版中增加了两章讲述应用和数学金融，传承之前版本详细、严谨的风格，讲述了有价证券和期货理论的基本知识。书中用最初等的方法讲述了概率测度、随机变量、分布以及期望等基本概念。离散和连续的案例都有所涉及，在讲述后者的时候运用了微积分知识。配以大量的典型例子重点讲述概率推理，集中介绍了组合问题、Poison过程、随机漫步、遗传模型和Markov链。每章末都附有习题及其解答。