Hybrid Switching Diffusions

Table of Contents

1. Frontmatter

2. Chapter 1. Introduction and Motivation

   Hai-Dang Nguyen, George Yin, Chao Zhu

   Abstract

   This chapter begins with motivations of the study, delineates switching diffusions in an intuitive way, presents a number of application examples, and gives an outline of the entire book.

3. Chapter 2. Switching Diffusion

   Hai-Dang Nguyen, George Yin, Chao Zhu

   This chapter provides a rigorous introduction to switching diffusions. After presenting the definition of switching diffusion, this chapter establishes the existence and uniqueness of solutions to the associated stochastic differential equations under non-Lipschitz conditions, where the switching component has a countable state space. It then proceeds to examine the key properties of switching diffusions such as weak continuity, Feller and strong Feller properties. Moreover, the definitions of regularity and corresponding criteria are discussed. Furthermore, smooth dependence on initial data is demonstrated.

4. Chapter 3. Recurrence

   Hai-Dang Nguyen, George Yin, Chao Zhu

   Abstract

   Chapter 3 is concerned with recurrence of switching diffusion processes. Enabling the Markov process returns to any compact set infinitely often with probability one, recurrence guarantees long-term stability by preventing the process from escaping to infinity and ensures the existence of an invariant measure under appropriate conditions. This chapter provides a systematic study on the recurrence of switching diffusion processes. It establishes sufficient conditions for recurrence, transience, positive recurrence, and null recurrence using appropriate Lyapunov functions. Furthermore, it provides easily verifiable conditions for positive recurrence of linearized systems.

5. Chapter 4. Ergodicity

   Hai-Dang Nguyen, George Yin, Chao Zhu

   Abstract

   Chapter 4 is concerned with ergodicity and reveals the main features of the ergodic measures. The study on ergodicity is carried out by constructing “cycles” and using the induced discrete-time Markov chains. Irreducibility of switching diffusion processes combined with the strong Feller property ensures the uniqueness of the invariant measure for the underlying processes. In addition, this chapter explores feedback controls for weak stabilization.

6. Chapter 5. Numerical Approximation

   Hai-Dang Nguyen, George Yin, Chao Zhu

   Abstract

   This chapter is devoted to numerical approximation methods for switching diffusions whose switching component is x dependent. As closed-form solutions for switching diffusions are often difficult to obtain, numerical approximation is frequently a viable or possibly the only alternative. This chapter introduces numerical algorithms and establishes weak convergence of the associated continuous-time interpolation process using the martingale method. In addition, under mild conditions, it provides an explicit strong convergence rate for the numerical scheme.

7. Chapter 6. Stability

   Hai-Dang Nguyen, George Yin, Chao Zhu

   Abstract

   This chapter provides another perspective of the long-time behavior of switching diffusions by focusing on stability of an equilibrium point. Suffcient conditions for stability and instability in probability, exponential p-stability and almost sure exponential stability are presented. Moreover, easily verifiable conditions for stability and instability of linearized systems are provided. The main machineries are Lyapunov function methods.

8. Chapter 7. Stability of Switching ODEs

   Hai-Dang Nguyen, George Yin, Chao Zhu

   Abstract

   This chapter deals with stability of deterministic differential equations modulated by a randomly switching process with motivation given by the study of piecewise deterministic systems and stability of randomly switched systems. Also considered are the Lyapunov exponent and related stability issues.

9. Chapter 8. Invariance Principles

   Hai-Dang Nguyen, George Yin, Chao Zhu

   Abstract

   This chapter is concerned with invariance principles of switching diffusion processes. Together with Chaps. 6 and 7, it delineates longtime behavior and gives a comprehensive understanding of the asymptotic properties of switching diffusion processes.

10. Chapter 9. Two-Time-Scale Switching Diffusions

    Hai-Dang Nguyen, George Yin, Chao Zhu

    Abstract

    This chapter focuses on switching diffusions with two-time scales. It consists of two parts. The first part considers a stochastic volatility model using regime-switching diffusions with fast mean reversion. It develops asymptotic expansions for option pricing and establishes the asymptotic error bounds for these expansions. The second part of the chapter considers the states of the discrete event process belonging to several “ergodic” classes that are weakly connected. The underlying switching diffusion is shown to be positive recurrent under mild conditions.

11. Chapter 10. Switching Jump Diffusions: Time-Scale Separations

    Hai-Dang Nguyen, George Yin, Chao Zhu

    Abstract

    This chapter is concerned with two-time scale jump diffusions modulated by a continuous-time Markov chain to reduce complexity through an appropriate averaged system. The first part treats switching jump diffusion with fast switching. It demonstrates that the original complex problem can be “replaced” by a limit problem in which the system coeffcients are averaged out with respect to the stationary measures of the switching process. The second part deals with switching jump diffusion models with periodic fast-varying diffusion, and establishes the weak convergence of the process and with explicitly characterization of the limit system. This chapter also discusses numerical solutions for switching jump diffusions.

12. Chapter 11. Past-Dependent Switching Diffusion

    Hai-Dang Nguyen, George Yin, Chao Zhu

    This chapter presents an in-depth study of switching diffusions with countable state spaces, focusing on the crucial aspect of history-dependent switching. Specifically, the switching mechanism depends on the past trajectory of the continuous states. This chapter begins by demonstrating the existence and uniqueness of solutions to the associated stochastic differential equations. Then it proceeds to examine fundamental properties of these processes such as Markov, Feller, strong Feller, recurrence, ergodicity, and stability.

13. Chapter 12. Population Dynamics Modeled by Switching Diffusion

    Hai-Dang Nguyen, George Yin, Chao Zhu

    This chapter explores population dynamics and infectious disease systems in random environments modeled by switching diffusions. It first develops a general theory for coexistence and persistence of population dynamics driven by switching diffusions, followed by in-depth investigations of a chemostat model and an infectious disease model with vaccination.

14. Backmatter