Regime-switching Diffusion Research Articles

Asymptotic expansions of option price under regime-switching diffusions with a fast-varying switching process

Asymptotic expansions of option price under regime-switching diffusions with a fast-varying switching process

Rates of Convergence of Numerical Methods for Controlled Regime-Switching Diffusions with Stopping Times in the Costs

This work is concerned with rates of convergence of Markov chain approximation methods for controlled switching diffusions. The cost function is defined on an infinite horizon with stopping times and without discount. Displaying both continuous dynamics and discrete events, the discrete events are modeled by continuous-time Markov chains to delineate a random environment and other random factors that cannot be represented by diffusion processes. This paper presents a first attempt using a probabilistic approach to treat such rates of convergence problems. In addition, in contrast to the significant developments in the literature using partial differential equation (PDE) methods for the approximation of controlled diffusions, there do not yet appear to be any PDE results to date for rates of convergence of numerical solutions for controlled switching diffusions, to the best of our knowledge. Although some of the working conditions in this paper such as the one-dimensional continuous state variable, nondegenerate diffusions, and control only on the drift may be seemingly strong, they are adequate as the starting point for using this new approach to treat the rates of convergence problems. Moreover, in the literature, to prove the convergence using Markov chain approximation methods for control problems involving cost functions with stopping (even for uncontrolled diffusion without switching), an added assumption was used to avoid the so-called tangency problem. As a by-product of our approach, by modifying the value function, it is demonstrated that the anticipated tangency problem will not arise in the sense of convergence in probability and in the sense of $L^1$.

On Strong Feller, Recurrence, and Weak Stabilization of Regime-Switching Diffusions

Focusing on two-component Markov processes having continuous dynamics and discrete events, this work develops asymptotic properties of regime-switching diffusions. First, strong Feller property is established. Next, classifying the underlying processes as recurrence (positive recurrence or null recurrence) or transience, sufficient conditions for such processes are obtained. In addition to providing general criteria for recurrent and transient switching diffusions, for processes that are linearizable with respect to the continuous component, easily verifiable conditions are established. Also provided are conditions for null-recurrent processes. Furthermore, conditions for controlled switching diffusions in which the feedback controls ensure weak stability (or ergodicity) are given. Finally, simple examples are used for demonstration.

Practical stability and instability of regime-switching diffusions

This work is devoted to practical stability of a class of regime-switching diffusions. First, the notion of practical stability is introduced. Then, sufficient conditions for practical stability and practical instability in probability and in pth mean are provided using a Lyapunov function argument. In addition, easily verifiable conditions on drift and diffusion coefficients are also given. Moreover, examples are supplied for demonstration purposes.

Numerical Solutions for Stochastic Differential Games With Regime Switching

This paper is concerned with numerical methods for stochastic differential games of regime-switching diffusions. Numerical methods using Markov chain approximation techniques are developed. A new proof of the existence of a saddle point for the stochastic differential game is provided. This new proof enables us to treat certain systems with nonseparable (in controls) structure. Convergence of the algorithms is derived by means of weak convergence methods. In addition, examples are also provided for demonstration purposes.

Regularity and Recurrence of Switching Diffusions

This work is concerned with switching diffusion processes, also known as regime-switching diffusions. Our attention focuses on regularity, recurrence, and positive recurrence of the underlying stochastic processes. The main effort is devoted to obtaining easily verifiable conditions for the aforementioned properties. Continuous-state-dependent jump processes are considered. First general criteria on regularity and recurrence using Liapunov functions are obtained. Then we focus on a class of problems, in which both the drift and the diffusion coefficients are “linearizable” with respect to the continuous state, and suppose that the generator of the jump part of the process can be approximated by a generator of an ergodic Markov chain. Sufficient conditions for regularity, recurrence, and positive recurrence are derived, which are linear combination of the averaged coefficients (averaged with respect to the stationary measure of the Markov chain).

Stability of regime-switching diffusions

This work is devoted to stability of regime-switching diffusion processes. After presenting the formulation of regime-switching diffusions, the notion of stability is recalled, and necessary conditions for p -stability are obtained. Then main results on stability and instability for systems arising in approximation are presented. Easily verifiable conditions are established. An example is examined as a demonstration. A remark on linear systems is also provided.

Asymptotic Properties of Hybrid Diffusion Systems

In response to the increasing needs for control and optimization of hybrid systems, this work is concerned with such asymptotic properties as recurrence (also known as weak stochastic stability in the literature) and ergodicity of regime-switching diffusions. Using Liapunov functions, necessary and sufficient conditions for positive recurrence are developed. Then, ergodicity of positive recurrent regime-switching diffusions is obtained by constructing cycles using the associated discrete-time Markov chains.

Numerical methods for controlled regime-switching diffusions and regime-switching jump diffusions

This work is concerned with numerical methods for controlled regime-switching diffusions, and regime-switching jump diffusions. Numerical procedures based on Markov chain approximation techniques are developed. Convergence of the algorithms is derived by means of weak convergence methods. In addition, examples are also provided for demonstration purpose.

Numerical approximation of invariant measures for hybrid diffusion systems

This work is concerned with numerical approximation of hybrid diffusions with regime switching modulated by continuous-time finite-state Markov chains. When using the Euler-Maruyama approximation algorithms, an important question is: Suppose the measure of the regime-switching diffusions converges to its invariant measure, will the sequence of measures of the approximation scheme converges to the same limit? This paper provides answers to this question. By appropriate interpolations and weak convergence methods, it shows that a suitably interpolated sequence resulted from the algorithm converges to the switching diffusion. The convergence to the invariant measure of the numerical algorithm is also studied.

Estimation of Continuous-Time Markov Processes Sampled at Random Time Intervals

We introduce a family of generalized-method-of-moments estimators of the parameters of a continuous-time Markov process observed at random time intervals. The results include strong consistency, asymptotic normality, and a characterization of standard errors. Sampling is at an arrival intensity that is allowed to depend on the underlying Markov process and on the parameter vector to be estimated. We focus on financial applications, including tick-based sampling, allowing for jump diffusions, regime-switching diffusions, and reflected diffusions.