文档介绍：Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Motivation: Stochastic Differential Equations . . . . . . . . . . . . . . . 1
The Obstacle 4, Itˆo’s Way Out of the Quandary 5, Summary: The Task Ahead 6
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
The General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Filtrations on Measurable Spaces 21, The Base Space 22, Processes 23, Stop-
ping 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
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
The Semivariations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
The Size of an Integrator 54, Vectors of Integrators 56, The Natural Conditions 56
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
Processes of Finite Variation . . . . .