Stochastic Integration with Jumps

Title page
Preface
Chapter 1 Introduction
1.1 Motivation: Stochastic Differential Equations
The Obstacle
Itô's Way Out of the Quandary
Summary: The Task Ahead
1.2 Wiener Process
Existence of Wiener Process
Uniqueness of Wiener Measure
Non-Differentiability of the Wiener Path
Supplements and Additional Exercises
1.3 The General Model
Filtrations on Measurable Spaces
The Base Space
Processes
Stopping Times and Stochastic Intervals
Some Examples of Stopping Times
Probabilities
The Sizes of Random Variables
Two Notions of Equality for Processes
The Natural Conditions
Chapter 2 Integrators and Martingales
Step Functions and Lebesgue-Stieltjes Integrators on the Line
2.1 The Elementary Stochastic Integral
Elementary Stochastic Integrands
The Elementary Stochastic Integral
The Elementary Integral and Stopping Times
L^p-Integrators
Local Properties
2.2 The Semivariations
The Size of an Integrator
Vectors of Integrators
The Natural Conditions
2.3 Path Regularity of Integrators
Right-Continuity and Left Limits
Boundedness of the Paths
Redefinition of Integrators
The Maximal Inequality
Law and Canonical Representation
2.4 Processes of Finite Variation
Decomposition into Continuous and Jump Parts
The Change-of-Variable Formula
2.5 Martingales
Submartingales and Supermartingales
Regularity of the Paths: Right-Continuity and Left Limits
Boundedness of the Paths
Doob's Optional Stopping Theorem
Martingales Are Integrators
Martingales in L^p
Chapter 3 Extension of the Integral
Daniell's Extension Procedure on the Line
3.1 The Daniell Mean
A Temporary Assumption
Properties of the Daniell Mean
3.2 The Integration Theory of a Mean
Negligible Functions and Sets
Processes Finite for the Mean and Defined Almost Everywhere
Integrable Processes and the Stochastic Integral
Permanence Properties of Integrable Functions
Permanence Under Algebraic and Order Operations
Permanence Under Pointwise Limits of Sequences
Integrable Sets
3.3 Countable Additivity in p-Mean
The Integration Theory of Vectors of Integrators
3.4 Measurability
Permanence Under Limits of Sequences
Permanence Under Algebraic and Order Operations
The Integrability Criterion
Measurable Sets
3.5 Predictable and Previsible Processes
Predictable Processes
Previsible Processes
Predictable Stopping Times
Accessible Stopping Times
3.6 Special Properties of Daniell's Mean
Maximality
Continuity Along Increasing Sequences
Predictable Envelopes
Regularity
Stability Under Change of Measure
3.7 The Indefinite Integral
The Indefinite Integral
Integration Theory of the Indefinite Integral
A General Integrability Criterion
Approximation of the Integral via Partitions
Pathwise Computation of the Indefinite Integral
Integrators of Finite Variation
3.8 Functions of Integrators
Square Bracket and Square Function of an Integrator
The Square Bracket of Two Integrators
The Square Bracket of an Indefinite Integral
Application: The Jump of an Indefinite Integral
3.9 Itô's Formula
The Doléans-Dade Exponential
Additional Exercises
Girsanov Theorems
The Stratonovich Integral
3.10 Random Measures
σ-Additivity
Law and Canonical Representation
Example: Wiener Random Measure
Example: The Jump Measure of an Integrator
Strict Random Measures and Point Processes
Example: Poisson Point Processes
The Girsanov Theorem for Poisson Point Processes
Chapter 4 Control of Integral and Integrator
4.1 Change of Measure - Factorization
A Simple Case
The Main Factorization Theorem
Proof for p > 0
Proof for p = 0
4.2 Martingale Inequalities
Fefferman's Inequality
The Burkholder-Davis-Gundy Inequalities
The Hardy Mean
Martingale Representation on Wiener Space
Additional Exercises
4.3 The Doob-Meyer Decomposition
Doléans-Dade Measures and Processes
Proof of Theorem 4.3.1: NecessityUniquenessand Existence
Proof of Theorem 4.3.1: The Inequalities
The Previsible Square Function
The Doob-Meyer Decomposition of a Random Measure
4.4 Semimartingales
Integrators Are Semimartingales
Various Decompositions of an Integrator
4.5 Previsible Control of Integrators
Controlling a Single Integrator
Previsible Control of Vectors of Integrators
Previsible Control of Random Measures
4.6 Lévy Processes
The Lévy-Khintchine Formula
The Martingale Representation Theorem
Canonical Components of a Lévy Process
Construction of Lévy Processes
Feller Semigroup and Generator
Chapter 5 Stochastic Differential Equations
5.1 Introduction
First Assumptions on the Data and Definition of Solution
Example: The Ordinary Differential Equation (ODE)
ODE: Flows and Actions
ODE: Approximation
5.2 Existence and Uniqueness of the Solution
The Picard Norms
Lipschitz Conditions
Existence and Uniqueness of the Solution
Stability
Differential Equations Driven by Random Measures
The Classical SDE
5.3 Stability: Differentiability in Parameters
The Derivative of the Solution
Pathwise Differentiability
Higher Order Derivatives
5.4 Pathwise Computation of the Solution
The Case of Markovian Coupling Coefficients
The Case of Endogenous Coupling Coefficients
The Univers al Solution
A Non-Adaptive Scheme
The Stratonovich Equation
Higher Order Approximation: Obstructions
Higher Order Approximation: Results
5.5 Weak Solutions
The Size of the Solution
Existence of Weak Solutions
Uniqueness
5.6 Stochastic Flows
Stochastic Flows with a Continuous Driver
Drivers with Small Jumps
Markovian Stochastic Flows
Markovian Stochastic Flows Driven by a Lévy Process
5.7 Semigroups, Markov Processes, and PDE
Stochastic Representation of Feller Semigroups
Appendix A Complements to Topology and Measure Theory
A.1 Notations and Conventions
A.2 Topological Miscellanea
The Theorem of Stone-Weierstrass
TopologiesFiltersUniformities
Semi-continuity
Separable Metric Spaces
Topological Vector Spaces
The Minimax Theorem, Lemmas of Gronwall and Kolmogoroff
Differentiation
A.3 Measure and Integration
σ-Algebras
Sequential Closure
Measures and Integrals
Order-Continuous and Tight Elementary Integrals
Projective Systems of Measures
Products of Elementary Integrals
Infinite Products of Elementary Integrals
Images, Law, and Distribution
The Vector Lattice of All Measures
Conditional Expectation
Numerical and σ-Finite Measures
Characteristic Functions
Convolution
Liftings, Disintegration of Measures
Gaussian and Poisson Random Variables
A.4 Weak Convergence of Measures
Uniform Tightness
Application: Donsker's Theorem
A.5 Analytic Sets and Capacity
Applications to Stochastic Analysis
Supplements and Additional Exercises
A.6 Suslin Spaces and Tightness of Measures
Polish and Suslin Spaces
A.7 The Skorohod Topology
A.8 The L^p Spaces
Marcinkiewicz Interpolation
Khintchine's Inequalities
Stable Type
A.9 Semigroups of Operators
Resolvent and Generator
Feller Semigroups
The Natural Extension of a Feller Semigroup
Appendix B Answers to Selected Problems
References
Index of Notations
Index
Answers http://www.ma.utexas.edu/users/cup/Answers Full Indexes http://www.ma.utexas.edu/users/cup/lndexes
Errata http://www.ma.utexas.edu/users/cup/Errata