斯坦福大学Amir Dembo的随机过程讲义

　　　目　　　　录

Chapter 1. Probability, measure and integration
1.1. Probability spaces and σ-fields
1.2. Random variables and their expectation
1.3. Convergence of random variables
1.4. Independence, weak convergence and uniform integrability

Chapter 2. Conditional expectation and Hilbert spaces
2.1. Conditional expectation: existence and uniqueness
2.2. Hilbert spaces
2.3. Properties of the conditional expectation
2.4. Regular conditional probability

Chapter 3. Stochastic Processes: general theory
3.1. Definition, distribution and versions
3.2. Characteristic functions, Gaussian variables and processes
3.3. Continuity, separability and measurability

Chapter 4. Martingales and stopping times
4.1. Discrete time martingales and filtrations
4.2. Continuous time martingales and right continuous filtrations
4.3. Stopping times and the optional stopping theorem
4.4. Martingale representations and inequalities
4.5. Martingales: convergence theorems and applications
4.6. Branching processes: extinction probabilities

Chapter 5. The Brownian motion
5.1. Brownian motion: definition and construction
5.2. The reflection principle and Brownian hitting times
5.3. Smoothness and variation of the Brownian sample path

Chapter 6. Markov, Poisson and Jump processes
6.1. Markov chains and processes
6.2. Poisson process, Exponential inter-arrivals and order statistics
6.3. Markov jump processes