Stochastic Differential Equations with Markovian Switching

The following sections are included:

• Preface

• Contents

• Notation

The following sections are included:

• Introduction

• Basic Notations of Probability Theory

• Stochastic Processes

• Brownian Motions

• Stochastic Integrals

• Itô's Formula

• Markov Processes

• Generalised Itô's Formula

• Exercises

The following sections are included:

• Introduction

• Frequently Used Inequalities

• Gronwall-Type Inequalities

• Matrices and Inequalities

• Linear Matrix Inequalities

• M-Matrix Inequalities

• Stochastic Inequalities

• Exercises

The following sections are included:

• Introduction

• Stochastic Differential Equations

• Existence and Uniqueness of Solutions

• SDEs with Markovian Switching

• L^p-Estimate

• Almost Surely Asymptotic Estimates

• Solutions as Markov Processes

• Exercises

The following sections are included:

• Introduction

• Euler–Maruyama's Approximations
  • Global Lipschitz Case
  • Local Lipschitz Case
  • More on Local Lipschitz Case

• Caratheodory's Approximations

• Split-Step Backward Euler Scheme

• Backward Euler Scheme

• Stochastic Theta Method

• Exercises

The following sections are included:

• Introduction

• Asymptotic Boundedness

• Exponential Stability
  • Nonlinear Jump Systems
  • Multi-Dimensional Linear Equations
  • Scalar Linear Equations
  • Examples

• Moment and Almost Sure Asymptotic Stability

• Stability in Probability

• Asymptotic Stability in Distribution

• Exercises

The following sections are included:

• Introduction

• Euler–Maruyama's Method and Exponential Stability

• Euler–Maruyama's Method and Lyapunov Exponents

• Generalised Results and Stochastic Theta Method

• Asymptotic Stability in Distribution of the EM Method: Constant Step Size
  • Stability in Distribution of the EM Method
  • Sufficient Criteria for Assumptions 6.16–6.18
  • Convergence of Stationary Distributions

• Asymptotic Stability in Distribution of the EM Method: Variable Step Sizes

• Exercises

The following sections are included:

• Introduction

• Stochastic Differential Delay Equations

• SDDEs with Markovian Switching

• Moment Properties

• Asymptotic Boundedness

• Exponential Stability

• Approximate Solutions

• Exercises

The following sections are included:

• Introduction

• Stochastic Functional Differential Equations

• SFDEs with Markovian Switching

• Boundedness

• Asymptotic Stability

• Razumikhin-Type Theorems on Stability

• Examples

• Exercises

The following sections are included:

• Introduction

• Interval Matrices

• SDISs with Markovian Switching

• Razumikhin Technology on SDISs
  • Delay Independent Criteria
  • Delay Dependent Criteria
  • Examples

• SISs with Markovian Switching

• Exercises

The following sections are included:

• Introduction

• Stochastic Population Dynamics
  • Global Positive Solutions
  • Ultimate Boundedness
  • Moment Average in Time

• Stochastic Financial Modelling
  • Non-Negative Solutions
  • The EM Approximations
  • Stochastic Volatility Model
  • Options Under Stochastic Volatility

• Stochastic Stabilisation and Destabilisation

• Stochastic Neural Networks

• Exercises

The following sections are included:

• Bibliographical Notes

• Bibliography

• Index