Contents
Preface 5
Chapter 1. Probability, measure and integration 7
1.1. Probability spaces and σ-fields 7
1.2. Random variables and their expectation 10
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 Pro-
cesses 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 ad-
vanced 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 the-
ory 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 σ-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 pro-
gression 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, conver-
gence 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 mo-
tion. 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
,6 PREFACE
Chapter 6 provides a brief introduction to the theory of Markov chains and pro-
cesses, a vast subject at the core of probability theory, to which many text books
are devoted. We illustrate some of the interesting mathematical properties of such
processes by examining the special case of the Poisson process, and more generally,
that of Markov jump processes.
As clear from the preceding, it normally takes more than a year to cover the scope
of this text. Even more so, given that the intended audience for this course has only
minimal prior exposure to stochastic processes (beyond the usual elementary prob-
ability class covering only discrete settings and variables with probability density
function). While students are assumed to have taken a real analysis class dealing
with Riemann integration, no prior knowledge of measure theory is assumed here.
The unusual solution to this set of constraints is to provide rigorous definitions,
examples and theorem statements, while forgoing the proofs of all but the most
easy derivations. At this somewhat superficial level, one can cover everything in a
one semester course of forty lecture hours (and if one has highly motivated students
such as I had in Stanford, even a one quarter course of thirty lecture hours might
work).
In preparing this text I was much influenced by Zakai’s unpublished lecture notes
[Zak]. Revised and expanded by Shwartz and Zeitouni it is used to this day for
teaching Electrical Engineering Phd students at the Technion, Israel. A second
source for this text is Breiman’s [Bre92], which was the intended text book for my
class in 2002, till I realized it would not do given the preceding constraints. The
resulting text is thus a mixture of these influencing factors with some digressions
and additions of my own.
I thank my students out of whose work this text materialized. Most notably I
thank Nageeb Ali, Ajar Ashyrkulova, Alessia Falsarone and Che-Lin Su who wrote
the first draft out of notes taken in class, Barney Hartman-Glaser, Michael He,
Chin-Lum Kwa and Chee-Hau Tan who used their own class notes a year later in
a major revision, reorganization and expansion of this draft, and Gary Huang and
Mary Tian who helped me with the intricacies of LATEX.
I am much indebted to my colleague Kevin Ross for providing many of the exercises
and all the figures in this text. Kevin’s detailed feedback on an earlier draft of these
notes has also been extremely helpful in improving the presentation of many key
concepts.
Amir Dembo
Stanford, California
January 2008
, CHAPTER 1
Probability, measure and integration
This chapter is devoted to the mathematical foundations of probability theory.
Section 1.1 introduces the basic measure theory framework, namely, the proba-
bility space and the σ-fields of events in it. The next building block are random
variables, introduced in Section 1.2 as measurable functions ω 7→ X(ω). This allows
us to define the important concept of expectation as the corresponding Lebesgue
integral, extending the horizon of our discussion beyond the special functions and
variables with density, to which elementary probability theory is limited. As much
of probability theory is about asymptotics, Section 1.3 deals with various notions
of convergence of random variables and the relations between them. Section 1.4
concludes the chapter by considering independence and distribution, the two funda-
mental aspects that differentiate probability from (general) measure theory, as well
as the related and highly useful technical tools of weak convergence and uniform
integrability.
1.1. Probability spaces and σ-fields
We shall define here the probability space (Ω, F , P) using the terminology of mea-
sure theory. The sample space Ω is a set of all possible outcomes ω ∈ Ω of some
random experiment or phenomenon. Probabilities are assigned by a set function
A 7→ P(A) to A in a subset F of all possible sets of outcomes. The event space F
represents both the amount of information available as a result of the experiment
conducted and the collection of all events of possible interest to us. A pleasant
mathematical framework results by imposing on F the structural conditions of a
σ-field, as done in Subsection 1.1.1. The most common and useful choices for this
σ-field are then explored in Subsection 1.1.2.
1.1.1. The probability space (Ω, F , P). We use 2Ω to denote the set of all
possible subsets of Ω. The event space is thus a subset F of 2Ω , consisting of all
allowed events, that is, those events to which we shall assign probabilities. We next
define the structural conditions imposed on F .
Definition 1.1.1. We say that F ⊆ 2Ω is a σ-field (or a σ-algebra), if
(a) Ω ∈ F,
(b) If A ∈ F then Ac ∈ F as well (where c
S∞A = Ω \ A).
(c) If Ai ∈ F for i = 1, 2 . . . then also i=1 Ai ∈ F.
Remark. Using DeMorgan’s T law you can easily check that if Ai ∈ F for i = 1, 2 . . .
and F is a σ-field, then also i Ai ∈ F. Similarly, you can show that a σ-field is
closed under countably many elementary set operations.
7
Preface 5
Chapter 1. Probability, measure and integration 7
1.1. Probability spaces and σ-fields 7
1.2. Random variables and their expectation 10
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 Pro-
cesses 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 ad-
vanced 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 the-
ory 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 σ-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 pro-
gression 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, conver-
gence 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 mo-
tion. 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
,6 PREFACE
Chapter 6 provides a brief introduction to the theory of Markov chains and pro-
cesses, a vast subject at the core of probability theory, to which many text books
are devoted. We illustrate some of the interesting mathematical properties of such
processes by examining the special case of the Poisson process, and more generally,
that of Markov jump processes.
As clear from the preceding, it normally takes more than a year to cover the scope
of this text. Even more so, given that the intended audience for this course has only
minimal prior exposure to stochastic processes (beyond the usual elementary prob-
ability class covering only discrete settings and variables with probability density
function). While students are assumed to have taken a real analysis class dealing
with Riemann integration, no prior knowledge of measure theory is assumed here.
The unusual solution to this set of constraints is to provide rigorous definitions,
examples and theorem statements, while forgoing the proofs of all but the most
easy derivations. At this somewhat superficial level, one can cover everything in a
one semester course of forty lecture hours (and if one has highly motivated students
such as I had in Stanford, even a one quarter course of thirty lecture hours might
work).
In preparing this text I was much influenced by Zakai’s unpublished lecture notes
[Zak]. Revised and expanded by Shwartz and Zeitouni it is used to this day for
teaching Electrical Engineering Phd students at the Technion, Israel. A second
source for this text is Breiman’s [Bre92], which was the intended text book for my
class in 2002, till I realized it would not do given the preceding constraints. The
resulting text is thus a mixture of these influencing factors with some digressions
and additions of my own.
I thank my students out of whose work this text materialized. Most notably I
thank Nageeb Ali, Ajar Ashyrkulova, Alessia Falsarone and Che-Lin Su who wrote
the first draft out of notes taken in class, Barney Hartman-Glaser, Michael He,
Chin-Lum Kwa and Chee-Hau Tan who used their own class notes a year later in
a major revision, reorganization and expansion of this draft, and Gary Huang and
Mary Tian who helped me with the intricacies of LATEX.
I am much indebted to my colleague Kevin Ross for providing many of the exercises
and all the figures in this text. Kevin’s detailed feedback on an earlier draft of these
notes has also been extremely helpful in improving the presentation of many key
concepts.
Amir Dembo
Stanford, California
January 2008
, CHAPTER 1
Probability, measure and integration
This chapter is devoted to the mathematical foundations of probability theory.
Section 1.1 introduces the basic measure theory framework, namely, the proba-
bility space and the σ-fields of events in it. The next building block are random
variables, introduced in Section 1.2 as measurable functions ω 7→ X(ω). This allows
us to define the important concept of expectation as the corresponding Lebesgue
integral, extending the horizon of our discussion beyond the special functions and
variables with density, to which elementary probability theory is limited. As much
of probability theory is about asymptotics, Section 1.3 deals with various notions
of convergence of random variables and the relations between them. Section 1.4
concludes the chapter by considering independence and distribution, the two funda-
mental aspects that differentiate probability from (general) measure theory, as well
as the related and highly useful technical tools of weak convergence and uniform
integrability.
1.1. Probability spaces and σ-fields
We shall define here the probability space (Ω, F , P) using the terminology of mea-
sure theory. The sample space Ω is a set of all possible outcomes ω ∈ Ω of some
random experiment or phenomenon. Probabilities are assigned by a set function
A 7→ P(A) to A in a subset F of all possible sets of outcomes. The event space F
represents both the amount of information available as a result of the experiment
conducted and the collection of all events of possible interest to us. A pleasant
mathematical framework results by imposing on F the structural conditions of a
σ-field, as done in Subsection 1.1.1. The most common and useful choices for this
σ-field are then explored in Subsection 1.1.2.
1.1.1. The probability space (Ω, F , P). We use 2Ω to denote the set of all
possible subsets of Ω. The event space is thus a subset F of 2Ω , consisting of all
allowed events, that is, those events to which we shall assign probabilities. We next
define the structural conditions imposed on F .
Definition 1.1.1. We say that F ⊆ 2Ω is a σ-field (or a σ-algebra), if
(a) Ω ∈ F,
(b) If A ∈ F then Ac ∈ F as well (where c
S∞A = Ω \ A).
(c) If Ai ∈ F for i = 1, 2 . . . then also i=1 Ai ∈ F.
Remark. Using DeMorgan’s T law you can easily check that if Ai ∈ F for i = 1, 2 . . .
and F is a σ-field, then also i Ai ∈ F. Similarly, you can show that a σ-field is
closed under countably many elementary set operations.
7
