Book:Jean-François Le Gall/Brownian Motion, Martingales, and Stochastic Calculus

Subject Matter

 * Probability Theory

Contents

 * 1 Events and their probabilities
 * 1.1 Introduction
 * 1.2 Events as sets
 * 1.3 Probability
 * 1.4 Conditional probability
 * 1.5 Independence
 * 1.6 Completeness and product spaces
 * 1.7 Worked examples
 * 1.8 Problems


 * 2 Random variables and their distributions
 * 2.1 Random variables
 * 2.2 The law of averages
 * 2.3 Discrete and continuous variables
 * 2.4 Worked examples
 * 2.5 Random vectors
 * 2.6 Monte Carlo simulation
 * 2.7 Problems


 * 3 Discrete random variables
 * 3.1 Probability mass functions
 * 3.2 Independence
 * 3.3 Expectation
 * 3.4 Indicators and matching
 * 3.5 Examples of discrete variables
 * 3.6 Dependence
 * 3.7 Conditional distributions and conditional expectation
 * 3.8 Sums of random variables
 * 3.9 Simple random walk
 * 3.10 Random walk: counting sample paths
 * 3.11 Problems


 * 4 Continuous random variables
 * 4.1 Probability density functions
 * 4.2 Independence
 * 4.3 Expectation
 * 4.4 Examples of continuous variables
 * 4.5 Dependence
 * 4.6 Conditional distributions and conditional expectation
 * 4.7 Functions of random variables
 * 4.8 Sums of random variables
 * 4.9 Multivariate normal distribution
 * 4.10 Distributions arising from the normal distribution
 * 4.11 Sampling from a distribution
 * 4.12 Coupling and Poisson approximation
 * 4.13 Geometrical probability
 * 4.14 Problems


 * 5 Generating functions and their applications
 * 5.1 Generating functions
 * 5.2 Some applications
 * 5.3 Random walk
 * 5.4 Branching processes
 * 5.5 Age-dependent branching processes
 * 5.6 Expectation revised
 * 5.7 Characteristic functions
 * 5.8 Examples of characteristic functions
 * 5.9 Inversion and continuity theorems
 * 5.10 Two limit theorems
 * 5.11 Large deviations
 * 5.12 Problems


 * 6 Markov chains
 * 6.1 Markov processes
 * 6.2 Classification of states
 * 6.3 Classification of chains
 * 6.4 Stationary distributions and the limit theorem
 * 6.5 Reversibility
 * 6.6 Chains with finitely many states
 * 6.7 Branching processes revisited
 * 6.8 Birth processes and the Poisson process
 * 6.9 Continuous-time Markov chains
 * 6.10 Uniform semigroups
 * 6.11 Birth-death processes and imbedding
 * 6.12 Special processes
 * 6.13 Spatial Poisson processes
 * 6.14 Markov chain Monte Carlo
 * 6.15 Problems


 * 7 Convergence of random variables
 * 7.1 Introduction
 * 7.2 Modes of convergence
 * 7.3 Some ancillary results
 * 7.4 Laws of large numbers
 * 7.5 The strong law
 * 7.6 The law of the iterated logarithm
 * 7.7 Martingales
 * 7.8 Martingale convergence theorem
 * 7.9 Prediction and conditional expectation
 * 7.10 Uniform integrability
 * 7.11 Problems


 * 8 Random processes
 * 8.1 Introduction
 * 8.2 Stationary processes
 * 8.3 Renewal processes
 * 8.4 Queues
 * 8.5 The Wiener process
 * 8.6 Existence of processes
 * 8.7 Problems


 * 9 Stationary processes
 * 9.1 Introduction
 * 9.2 Linear prediction
 * 9.3 Autocovariances and spectra
 * 9.4 Stochastic integration and the spectral representation
 * 9.5 The ergodic theorem
 * 9.6 Gaussian processes
 * 9.7 Problems


 * 10 Renewals
 * 10.1 The renewal equation
 * 10.2 Limit theorems
 * 10.3 Excess life
 * 10.4 Applications
 * 10.5 Renewal-reward processes
 * 10.6 Problems


 * 11 Queues
 * 11.1 Single-server queues
 * 11.2 M/M/1
 * 11.3 M/G/1
 * 11.4 G/M/1
 * 11.5 G/G/1
 * 11.6 Heavy traffic
 * 11.7 Networks of queues
 * 11.8 Problems


 * 12 Martingales
 * 12.1 Introduction
 * 12.2 Martingale differences and Hoeffding's inequality
 * 12.3 Crossings and convergence
 * 12.4 Stopping times
 * 12.5 Optional stopping
 * 12.6 The maximal inequality
 * 12.7 Backward martingales and continuous-time martingales
 * 12.8 Some examples
 * 12.9 Problems


 * 13 Diffusion processes
 * 13.1 Introduction
 * 13.2 Brownian motion
 * 13.3 Diffusion processes
 * 13.4 First passage times
 * 13.5 Barriers
 * 13.6 Excursions and the Brownian bridge
 * 13.7 Stochastic calculus
 * 13.8 The Itô integral
 * 13.9 Itô's formula
 * 13.10 Option pricing
 * 13.11 Passage probabilities and potentials
 * 13.12 Problems


 * Appendix I. Foundations and notation
 * Appendix II. Further reading
 * Appendix III. History and varieties of probability
 * Appendix IV. John Arbuthnot's Preface to Of the laws of chance (1692)
 * Appendix V. Table of distributions
 * Appendix VI. Chronology
 * Bibliography
 * Notation
 * Index