目錄
Preface
Introduction
Chapter 1.Probability Spaces
1.1.Sets and Sigma-Fields
1.2.Elementary Properties of Probability Spaces
1.3.The Intuition
1.4.Conditional Probability
1.5.Independence
1.6.Counting: Permutations and Combinations
1.7.The Gambler's Ruin
Chapter 2.Random Variables
2.1.Random Variables and Distributions
2.2.Existence of Random Variables
2.3.Independence of Random Variables
2.4.Types of Distributions
2.5.Expectations Ⅰ: Discrete Random Variables osodires
2.6.Moments, Means and Variances
2.7.Mean, Median, and Mode
2.8.Special Discrete Distributions
Chapter 3.Expectations Ⅱ: The General Case baв uundiiup8
3.1.From Discrete to Continuous
3.2.The Expectation as an Integral
3.3.Some Moment Inequalities
3.4.Convex Functions and Jensen's Inequality
3.5.Special Continuous Distributions
3.6.Joint Distributions and Joint Densities
3.7.Conditional Distributions, Densities, and Expectations
Chapter 4.Convergence
4.1.Convergence of Random Variables
4.2.Convergence Theorems for Expectations
4.3.Applications
Chapter 5.Laws of Large Numbers
5.1.The Weak and Strong Laws
5.2.Normal Numbers
5.3.Sequences of Random Variables: Existence*
5.4.Sigma Fields as Information
5.5.Another Look at Independence
5.6.Zero-one Laws
Chapter 6.Convergence in Distribution and the CLT
6.1.Characteristic Functions
6.2.Convergence in Distribution
6.3.Levy's Continuity Theorem
6.4.The Central Limit Theorem
6.5.Stable Laws*
Chapter 7.Markov Chains and Random Walks
7.1.Stochastic Processes
7.2.Markov Chains
7.3.Classification of States
7.4.Stopping Times
7.5.The Strong Markov Property
7.6.Recurrence and Transienc
7.7.Equilibrium and the Ergodic Theorem for Markov Chains
7.8.Finite State Markov Chains
7.9.Branching Processes
7.10.The Poisson Process
7.11.Birth and Death Processes*
Chapter 8.Conditional Expectations
8.1.Conditional Expectations
8.2.Elementary Properties
8.3.Approximations and Projections
Chapter 9.Discrete-Parameter Martingales
9.1.Martingales
9.2.System Theorems
9.3.Convergence
9.4.Uniform Integrability
9.5.Applications
9.6.Financial Mathematics Ⅰ: The Martingale Connection*
Chapter 10.Brownian Motion
10.1.Standard Brownian Motion
10.2.Stopping Times and the Strong Markov Property
10.3.The Zero Set of Brownian Motion
10.4.The Reflection Principle
10.5.Recurrence and Hitting Properties
10.6.Path Irregularity
10.7.The Brownian Infinitesimal Generator*
10.8.Related Processes
10.9.Higher Dimensional Brownian Motion
10.10.Financial Mathematics Ⅱ: The Black-Scholes Model*
10.11.Skorokhod Embedding*
10.12.Levy's Construction of Brownian Motion*
10.13.The Ornstein-Uhlenbeck Process*
10.14.White Noise and the Wiener Integral*
10.15.Physical Brownian Motion*
10.16.What Brownian Motion Really Does
Bibliography
Index
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