Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation: Stochastic Differential Equations . . . . . . . . . . . . . . . 1
The Obstacle 4, Itˆo’s Way Out of the Quandary 5, Summary: The Task Ahead 6
1.2 Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Existence of Wiener Process 11, Uniqueness of Wiener Measure 14, Non-
Differentiability of the Wiener Path 17, Supplements and Additional Exercises 18
1.3 The General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Filtrations on Measurable Spaces 21, The Base Space 22, Processes 23, Stopping
Times and Stochastic Intervals 27, Some Examples of Stopping Times 29,
Probabilities 32, The Sizes of Random Variables 33, Two Notions of Equality for
Processes 34, The Natural Conditions 36
Chapter 2 Integrators and Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Step Functions and Lebesgue–Stieltjes Integrators on the Line 43
2.1 The Elementary Stochastic Integral . . . . . . . . . . . . . . . . . . . . . . . . 46
Elementary Stochastic Integrands 46, The Elementary Stochastic Integral 47, The
Elementary Integral and Stopping Times 47, Lp -Integrators 49, Local Properties 51
2.2 The Semivariations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
The Size of an Integrator 54, Vectors of Integrators 56, The Natural Conditions 56
2.3 Path Regularity of Integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Right-Continuity and Left Limits 58, Boundedness of the Paths 61, Redefinition of
Integrators 62, The Maximal Inequality 63, Law and Canonical Representation 64
2.4 Processes of Finite Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Decomposition into Continuous and Jump Parts 69, The Change-of-Variable
Formula 70
2.5 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Submartingales and Supermartingales 73, Regularity of the Paths: Right-
Continuity and Left Limits 74, Boundedness of the Paths 76, Doob’s Optional
Stopping Theorem 77, Martingales Are Integrators 78, Martingales in Lp 80
Chapter 3 Extension of the Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Daniell’s Extension Procedure on the Line 87
3.1 The Daniell Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
A Temporary Assumption 89, Properties of the Daniell Mean 90
3.2 The Integration Theory of a Mean . . . . . . . . . . . . . . . . . . . . . . . . . 94
Negligible Functions and Sets 95, Processes Finite for the Mean and Defined Almost
Everywhere 97, Integrable Processes and the Stochastic Integral 99, Permanence
Properties of Integrable Functions 101, Permanence Under Algebraic and Order
Operations 101, Permanence Under Pointwise Limits of Sequences 102, Integrable