文档介绍：With Stochastic Proterieatd lm In-
Doob Polya
Kolmogorov Cramer
Borel Levy
Keynes Feller
Contents
PREFACE TO THE FOURTH EDITION xi
PROLOGUE TO INTRODUCTION TO
MATHEMATICAL FINANCE xiii
1SET 1
Sample sets 1
Operations with sets 3
Various relations 7
Indicator 13
Exercises 17
2 PROBABILITY 20
Examples of probability 20
Definition and illustrations 24
Deductions from the axioms 31
Independent events 35
Arithmetical density 39
Exercises 42
3COUNTING 46
Fundamental rule 46
Diverse ways of sampling 49
Allocation models; binomial coefficients 55
How to solve it 62
Exercises 70
vii
viii Contents
4 RANDOM VARIABLES 74
What is a random variable? 74
How do random e about? 78
Distribution and expectation 84
Integer-valued random variables 90
Random variables with densities 95
General case 105
Exercises 109
APPENDIX 1: BOREL FIELDS AND GENERAL
RANDOM VARIABLES 115
5CONDITIONING AND INDEPENDENCE 117
Examples of conditioning 117
Basic formulas 122
Sequential sampling 131
P´olya’s urn scheme 136
Independence and relevance 141
ical models 152
Exercises 157
6 MEAN, VARIANCE, AND TRANSFORMS 164
Basic properties of expectation 164
The density case 169
Multiplication theorem; variance and covariance 173
Multinomial distribution 180
Generating function and the like 187
Exercises 195
7 POISSON AND NORMAL DISTRIBUTIONS 203
Models for Poisson distribution 203
Poisson process 211
From binomial to normal 222
Normal distribution 229
Central limit theorem 233
Law of large numbers 239
Exercises 246
APPENDIX 2: STIRLING’S FORMULA AND
DE MOIVRE–LAPLACE’S THEOREM 251
Contents ix
8 FROM RANDOM WALKS TO MARKOV CHAINS 254
Problems of the wanderer or gambler 254
Limiting schemes 261
Transition probabilities 266
Basic structu