Efnisyfirlit

• Contents
• Preface to the First Edition
• Preface to the Second Edition
• 1 The need for measure theory
• 1.1 Various kinds of random variables
• 1.2 The uniform distribution and non-measurable sets
• 1.3 Exercises
• 1.4 Section summary
• 2 Probability triples
• 2.1 Basic definition
• 2.2 Constructing probability triples
• 2.3 The Extension Theorem
• 2.4 Constructing the Uniform[0 1] distribution
• 2.5 Extensions of the Extension Theorem
• 2.6 Coin tossing and other measures
• 2.7 Exercises
• 2.8 Section summary
• 3 Further probabilistic foundations
• 3.1 Random variables
• 3.2 Independence
• 3.3 Continuity of probabilities
• 3.4 Limit events
• 3.5 Tail fields
• 3.6 Exercises
• 3.7 Section summary
• 4 Expected values
• 4.1 Simple random variables
• 4.2 General non-negative random variables
• 4.3 Arbitrary random variables
• 4.4 The integration connection
• 4.5 Exercises
• 4.6 Section summary
• 5 Inequalities and convergence
• 5.1 Various inequalities
• 5.2 Convergence of random variables
• 5.3 Laws of large numbers
• 5.4 Eliminating the moment conditions
• 5.5 Exercises
• 5.6 Section summary
• 6 Distributions of random variables
• 6.1 Change of variable theorem
• 6.2 Examples of distributions
• 6.3 Exercises
• 6.4 Section summary
• 7 Stochastic processes and gambling games
• 7.1 A first existence theorem
• 7.2 Gambling and gambler’s ruin
• 7.3 Gambling policies
• 7.4 Exercises
• 7.5 Section summary
• 8 Discrete Markov chains
• 8.1 A Markov chain existence theorem
• 8.2 Transience recurrence and irreducibility
• 8.3 Stationary distributions and convergence
• 8.4 Existence of stationary distributions
• 8.5 Exercises
• 8.6 Section summary
• 9 More probability theorems
• 9.1 Limit theorems
• 9.2 Differentiation of expectation
• 9.3 Moment generating functions and large deviations
• 9.4 Fubini’s Theorem and convolution
• 9.5 Exercises
• 9.6 Section summary
• 10 Weak convergence
• 10.1 Equivalences of weak convergence
• 10.2 Connections to other convergence
• 10.3 Exercises
• 10.4 Section summary
• 11 Characteristic functions
• 11.1 The continuity theorem
• 11.2 The Central Limit Theorem
• 11.3 Generalisations of the Central Limit Theorem
• 11.4 Method of moments
• 11.5 Exercises
• 11.6 Section summary
• 12 Decomposition of probability laws
• 12.1 Lebesgue and Hahn decompositions
• 12.2 Decomposition with general measures
• 12.3 Exercises
• 12.4 Section summary
• 13 Conditional probability and expectation
• 13.1 Conditioning on a random variable
• 13.2 Conditioning on a sub-o-algebra
• 13.3 Conditional variance
• 13.4 Exercises
• 13.5 Section summary
• 14 Martingales
• 14.1 Stopping times
• 14.2 Martingale convergence
• 14.3 Maximal inequality
• 14.4 Exercises
• 14.5 Section summary
• 15 General stochastic processes
• 15.1 Kolmogorov Existence Theorem
• 15.2 Markov chains on general state spaces
• 15.3 Continuous-time Markov processes
• 15.4 Brownian motion as a limit
• 15.5 Existence of Brownian motion
• 15.6 Diffusions and stochastic integrals
• 15.7 Ito’s Lemma
• 15.8 The Black-Scholes equation
• 15.9 Section summary
• A. Mathematical Background
• A.1 Sets and functions
• A.2 Countable sets
• A.3 Epsilons and Limits
• A.4 Infimums and supremums
• A.5 Equivalence relations
• B. Bibliography
• B.1 Background in real analysis
• B.2 Undergraduate-level probability
• B.3 Graduate-level probability
• B.4 Pure measure theory
• B.5 Stochastic processes
• B.6 Mathematical finance
• Index