Chung K.L., AitSahlia F. Elementary Probability Theory: With Stochastic Processes and an Introduction to Mathematical Finance

Springer, 2010. — 402 p.In this edition two new chapters, 9 and 10, on mathematical finance are added. They are written by Dr. Farid AitSahlia, ancien eleve, 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.Preface to the Fourth EditionPrologue to Introduction to Mathematical FinanceSet
Sample sets
Operations with sets
Various relations
Indicator
ExercisesProbability
Examples of probability
Definition and illustrations
Deductions from the axioms
Independent events
Arithmetical density
ExercisesCounting
Fundamental rule
Diverse ways of sampling
Allocation models; binomial coefficients
How to solve it
ExercisesRandom Variables
What is a random variable?
How do random variables come about?
Distribution and expectation
Integer-valued random variables
Random variables with densities
General case
Exercises
Appendix 1 Borel Fields and General Random VariablesConditioning and Independence
Examples of conditioning
Basic formulas
Sequential sampling
P´olya’s urn scheme
Independence and relevance
Genetical models
ExercisesMean; Variance; and Transforms
Basic properties of expectation
The density case
Multiplication theorem; variance and covariance
Multinomial distribution
Generating function and the like
ExercisesPoisson and Normal Distributions
Models for Poisson distribution
Poisson process
From binomial to normal
Normal distribution
Central limit theorem
Law of large numbers
Exercises
Appendix 2 Stirling’s Formula and De Moivre–Laplace’s TheoremFrom Random Walks to Markov Chains
Problems of the wanderer or gambler
Limiting schemes
Transition probabilities
Basic structure of Markov chains
Further developments
Steady state
Winding up (or down?)
Exercises
Appendix 3 MartingaleMean-Variance Pricing Model
An investments primer
Asset return and risk
Portfolio allocation
Diversification
Mean-variance optimization
Asset return distributions
Stable probability distributions
Exercises
Appendix 4 Pareto and Stable LawsOption Pricing Theory
Options basics
Arbitrage-free pricing: 1-period model
Arbitrage-free pricing: N-period model
Fundamental asset pricing theorems
ExercisesGeneral ReferencesAnswers to ProblemsValues of the Standard Normal Distribution Function