Front Cover
Random Differential Inqualities
Copyright Page
Contents
Preface
Notations and Abbreviations
CHAPTER 1. Preliminary Analysis
1.0 Introduction
1.1 Events and Probability Measure
1.2 Random Variables, Distribution Functions, and Expectations
1.3 Convergence of Random Sequences
1.4 Conditional Probabilities and Expectations
1.5 Random Processes
1.6 Separability of Random Processes
1.7 Deterministic Comparison Theorems
Notes
CHAPTER 2. Sample Calculus Approach
2.0 Introduction
2.1 Sample Calculus
2.2 Existence and Continuation
2.3 Random Differential Inequalities
2.4 Maximal and Minimal Solutions
2.5 Random Comparison Principle
2.6 Uniqueness and Continuous Dependence
2.7 The Method of Variation of Parameters
2.8 Random Lyapunov Functions
2.9 Scope of Comparison Principle
2.10 Stability Concepts
2.11 Stability in Probability
2.12 Stability with Probability One
2.13 Stability in the pth Mean
Notes
CHAPTER 3. Lp-calculus Approach
3.0 Introduction
3.1 Lp-Calculus
3.2 Interrelationships between Sample and LP-Solutions
3.3 Existence and Uniqueness
3.4 Continuous Dependence
3.5 Comparison Theorems
3.6 Stability Criteria
Notes
CHAPTER 4. ItΓ΄-Doob Calculus Approach
4.0 Introduction
4.1 ItΓ΄βs Calculus
4.2 Existence and Uniqueness
4.3 Continuous Dependence
4.4 The Method of Variation of Parameters
4.5 Stochastic Differential Inequalities
4.6 Maximal and Minimal Solutions
4.7 Comparison Theorems
4.8 Lyapunov-Like Functions
4.9 Stability in Probability
4.10 Stability in the pth Mean
4.11 Stability with Probability One
Notes
Appendix
A.0. Introduction
A.1 Moments of Random Functions
A.2 Spectral Representations of Covariance and Correlation Functions
A.3 Some Properties of Gaussian Processes
A.4 Brownian Motion
A.5 Martingales
A.6 Metrically Transitive Processes
A.7 Markov Processes
A.8 Closed Graph Theorem
Notes
References
Index