Stochastic Processes - Stanford University

Stochastic Processes

Amir Dembo (revised by Kevin Ross) April 12, 2021

E-mail address: adembo@stanford.edu Department of Statistics, Stanford University, Stanford, CA 94305.

Contents

Preface

5

Chapter 1. Probability, measure and integration

7

1.1. Probability spaces and -fields

7

1.2. Random variables and their expectation

11

1.3. Convergence of random variables

19

1.4. Independence, weak convergence and uniform integrability

25

Chapter 2. Conditional expectation and Hilbert spaces

35

2.1. Conditional expectation: existence and uniqueness

35

2.2. Hilbert spaces

39

2.3. Properties of the conditional expectation

43

2.4. Regular conditional probability

46

Chapter 3. Stochastic Processes: general theory

49

3.1. Definition, distribution and versions

49

3.2. Characteristic functions, Gaussian variables and processes

55

3.3. Sample path continuity

62

Chapter 4. Martingales and stopping times

67

4.1. Discrete time martingales and filtrations

67

4.2. Continuous time martingales and right continuous filtrations

73

4.3. Stopping times and the optional stopping theorem

76

4.4. Martingale representations and inequalities

82

4.5. Martingale convergence theorems

88

4.6. Branching processes: extinction probabilities

90

Chapter 5. The Brownian motion

95

5.1. Brownian motion: definition and construction

95

5.2. The reflection principle and Brownian hitting times

101

5.3. Smoothness and variation of the Brownian sample path

103

Chapter 6. Markov, Poisson and Jump processes

111

6.1. Markov chains and processes

111

6.2. Poisson process, Exponential inter-arrivals and order statistics

119

6.3. Markov jump processes, compound Poisson processes

125

Bibliography

127

Index

129

3

Preface

These are the lecture notes for a one quarter graduate course in Stochastic Processes that I taught at Stanford University in 2002 and 2003. This course is intended for incoming master students in Stanford's Financial Mathematics program, for advanced undergraduates majoring in mathematics and for graduate students from Engineering, Economics, Statistics or the Business school. One purpose of this text is to prepare students to a rigorous study of Stochastic Differential Equations. More broadly, its goal is to help the reader understand the basic concepts of measure theory that are relevant to the mathematical theory of probability and how they apply to the rigorous construction of the most fundamental classes of stochastic processes.

Towards this goal, we introduce in Chapter 1 the relevant elements from measure and integration theory, namely, the probability space and the -fields of events in it, random variables viewed as measurable functions, their expectation as the corresponding Lebesgue integral, independence, distribution and various notions of convergence. This is supplemented in Chapter 2 by the study of the conditional expectation, viewed as a random variable defined via the theory of orthogonal projections in Hilbert spaces.

After this exploration of the foundations of Probability Theory, we turn in Chapter 3 to the general theory of Stochastic Processes, with an eye towards processes indexed by continuous time parameter such as the Brownian motion of Chapter 5 and the Markov jump processes of Chapter 6. Having this in mind, Chapter 3 is about the finite dimensional distributions and their relation to sample path continuity. Along the way we also introduce the concepts of stationary and Gaussian stochastic processes.

Chapter 4 deals with filtrations, the mathematical notion of information progression in time, and with the associated collection of stochastic processes called martingales. We treat both discrete and continuous time settings, emphasizing the importance of right-continuity of the sample path and filtration in the latter case. Martingale representations are explored, as well as maximal inequalities, convergence theorems and applications to the study of stopping times and to extinction of branching processes.

Chapter 5 provides an introduction to the beautiful theory of the Brownian motion. It is rigorously constructed here via Hilbert space theory and shown to be a Gaussian martingale process of stationary independent increments, with continuous sample path and possessing the strong Markov property. Few of the many explicit computations known for this process are also demonstrated, mostly in the context of hitting times, running maxima and sample path smoothness and regularity.

5

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