Regime-switching Diffusion Processes Research Articles

Comparison theorem and stability under perturbation of transition rate matrices for regime-switching processes

AbstractA comparison theorem for state-dependent regime-switching diffusion processes is established, which enables us to pathwise-control the evolution of the state-dependent switching component simply by Markov chains. Moreover, a sharp estimate on the stability of Markovian regime-switching processes under the perturbation of transition rate matrices is provided. Our approach is based on elaborate constructions of switching processes in the spirit of Skorokhod’s representation theorem varying according to the problem being dealt with. In particular, this method can cope with switching processes in an infinite state space and not necessarily of birth–death type. As an application, some known results on the ergodicity and stability of state-dependent regime-switching processes can be improved.

On subgeometric ergodicity of regime-switching diffusion processes

In this article, we discuss subgeometric ergodicity of a class of regime-switching diffusion processes. We derive conditions on the drift and diffusion coefficients, and the switching mechanism which result in subgeometric ergodicity of the corresponding semigroup with respect to the total variation distance as well as a class of Wasserstein distances. At the end, subgeometric ergodicity of certain classes of regime-switching Markov processes with jumps is also discussed.

Exponential ergodicity for regime-switching diffusion processes in total variation norm

<p style='text-indent:20px;'>We investigate the long time behavior for a class of regime-switching diffusion processes. Based on direct evaluation of moments and exponential functionals of hitting time of the underlying process, we adopt coupling method to obtain existence and uniqueness of the invariant probability measure and establish explicit exponential bounds for the rate of convergence to the invariant probability measure in total variation norm. In addition, we provide some concrete examples to illustrate our main results which reveal impact of random switching on stochastic stability and convergence rate of the system.</p>

Valuing Equity-Linked Guaranteed Minimum Death Benefits with European-Style Asian Payoffs Under the Regime-Switching Diffusion Process with Jump

Valuing Equity-Linked Guaranteed Minimum Death Benefits with European-Style Asian Payoffs Under the Regime-Switching Diffusion Process with Jump

Harnack Inequality and Long Time Asymptotics of Unbounded Additive Functionals of Regime-Switching Diffusion Processes

This work provides explicit conditions to establish the dimension-free Harnack inequalities respectively for state-independent and state-dependent regime-switching processes. Then such Harnack inequalities are applied to establish the hypercontractivity of the associated semigroup, and further to study the long time asymptotics of unbounded additive functionals of regime-switching processes. Much effort is devoted to overcoming the difficulty caused by the close interaction between the diffusion processes and the Markov chains in the setting of regime-switching processes.

Convergence, boundedness, and ergodicity of regime-switching diffusion processes with infinite memory

We study a class of diffusion processes, which are determined by solutions X(t) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain Λ(t). Under suitable conditions, we investigate convergence and boundedness of both the solutions X(t) and the functional solutions Xt. We show that two solutions (resp., functional solutions) from different initial data living in the same initial switching regime will be close with high probability as time variable tends to infinity, and that the solutions (resp., functional solutions) are uniformly bounded in the mean square sense. Moreover, we prove existence and uniqueness of the invariant probability measure of two-component Markov-Feller process (Xt, Λ(t)), and establish exponential bounds on the rate of convergence to the invariant probability measure under Wasserstein distance. Finally, we provide a concrete example to illustrate our main results.

Permanence and extinction of stochastic regime-switching mutualism model

This paper is concerned with the permanence and extinction of a stochastic regime-switching mutualism model. We aim to find the difference between the stochastic mutualism model with regime-switching and without regime-switching. By studying ergodicity of regime-switching diffusion processes, we establish the sufficient conditions to estimate the permanence and extinction of a species in a random switching environment. Moreover, compared with the system without switching, the advantages of the stochastic regime-switching mutualism model are given.

Regime-switching diffusion processes: strong solutions and strong Feller property

We investigate the existence and uniqueness of strong solutions for state-dependent regime-switching diffusion processes in an infinite state space with singular coefficients. Non-explosion conditions are given by using the Zvonkin’s transformation. The strong Feller property is proved by further assuming that the diffusion in each fixed environment generates a strong Feller semigroup, and our results can also be applied to irregular or degenerate situations.

A lattice-based approach to option and bond valuation under mean-reverting regime-switching diffusion processes

Nowadays, the pricing of financial instruments under continuous-time Markov switching models have received a widespread attention from researchers and practitioners in the finance industry. Lattice-based approaches are amongst the most widely used approaches to solve the pricing problem in this context. Recently, Yuen and Yang (2010) have proposed a simple and fast recombining trinomial tree method to handle the case of regime-switching geometric Brownian motion processes. In this paper, we generalize their approach to the regime-switching (exponential) mean-reverting case with state-dependent switching rates and derive the necessary conditions for the positivity of conditional branching probabilities. We employ the Hull and White’s tree-building procedure to limit tree growth away from the long-run mean of the process. We use the proposed lattice framework to price contingent claims of European, American and barrier type, and demonstrate its applicability in pricing default-free zero-coupon bonds. In the proposed tree structure, the number of nodes is substantially decreased and the computational cost is effectively reduced compared to the usual approaches. Extensive numerical experiments illustrate the efficiency and flexibility of the proposed scheme.

On invariant probability measures of regime-switching diffusion processes with singular drifts

Under integrability conditions of the coefficients, the existence, uniqueness and regularity estimates are derived for the invariant probability measure of regime-switching diffusion processes. Moreover, we study the L1-uniqueness of the semigroup generated by regime-switching diffusion processes. As an application, the L1-uniqueness of the extension of weakly coupled elliptic systems is obtained.

On pricing barrier control in a regime-switching regulated market

We study a pricing barrier control problem in a regime-switching regulated market. In doing so, we analyze a class of one-dimensional reflected regime-switching diffusion processes. Such diffusion models arise as the key approximating processes in a regulated financial market system with the presence of regime changes. Our main goal is to determine optimal pricing barriers as solutions of long-run average mean–variance optimization problems. More precisely, the optimal barrier, if exists, will be to maximize the long-run average expected return (i.e. steady-state mean) subject to a selected level of long-run average risk (i.e. steady-state variance).

Variational formula for the stability of regime-switching diffusion processes

The asymptotical stability in probability is studied for diffusion processes and regime-switching diffusion processes in this work. For diffusion processes, some criteria based on the integrability of the functionals of the coefficients are given, which yield a useful comparison theorem on stability with respect to some nonlinear systems. For regime-switching diffusion processes, some criteria based on the idea of a variational formula are given. Both state-independent and state-dependent regime-switching diffusion processes are investigated in this work. These conditions are easily verified and are shown to be sharp by examples.

Invariant Measures and Euler--Maruyama's Approximations of State-Dependent Regime-Switching Diffusions

Regime-switching processes contain two components: continuous component and discrete component, which can be used to describe a continuous dynamical system in a random environment. Such processes have many different properties other than general diffusion processes, and many more difficulties are needed to be overcome due to the intensive interaction between continuous and discrete components. In this work we give conditions for the existence and uniqueness of invariant measures for state-dependent regime-switching diffusion processes. Also, the strong convergence in the $L^1$-norm of a numerical approximation is established and its convergence rate is provided. A refined application of Skorokhod's representation of jumping processes plays a substantial role in this work.

On singular control problems with state constraints and regime-switching: A viscosity solution approach

This paper investigates a singular stochastic control problem for a multi-dimensional regime-switching diffusion process confined in an unbounded domain. The objective is to maximize the total expected discounted rewards from exerting the singular control. Such a formulation stems from application areas such as optimal harvesting multiple species and optimal dividends payments schemes in random environments. With the aid of weak dynamic programming principle and an exponential transformation, we characterize the value function to be the unique constrained viscosity solution of a certain system of coupled nonlinear quasi-variational inequalities. Several examples are analyzed in detail to demonstrate the main results.

Permanence and Extinction of Regime-Switching Predator-Prey Models

In this work we study the permanence and extinction of a regime-switching predator-prey model with Beddington-DeAngelis functional response. The switching process is used to describe the random changing of corresponding parameters such as birth and death rates of a species in different environments. Our criteria can justify whether a prey die out or not when it will die out in some environments and will not in others. Our criteria are rather sharp, and they cover the known on-off type results on permanence of predator-prey models without switching. Our method relies on the recent study of ergodicity of regime-switching diffusion processes.

Approximation of Invariant Measures for Regime-switching Diffusions

In this paper, we are concerned with long-time behavior of Euler-Maruyama schemes associated with a range of regime-switching diffusion processes. The key contributions of this paper lie in that existence and uniqueness of numerical invariant measures are addressed (i) for regime-switching diffusion processes with finite state spaces by the Perron-Frobenius theorem if the "averaging condition" holds, and, for the case of reversible Markov chain, via the principal eigenvalue approach provided that the principal eigenvalue is positive; (ii) for regime-switching diffusion processes with countable state spaces by means of a finite partition method and an M-Matrix theory. We also reveal that numerical invariant measures converge in the Wasserstein metric to the underlying ones. Several examples are constructed to demonstrate our theory.

Pricing dynamic fund protections with regime switching

This paper deals with the valuation of dynamic fund protections in a Markov regime-switching environment. The volatility switches over time subject to a continuous-time Markov chain. Using a regime-switching diffusion process to describe the primary mutual fund value, explicit solutions of the Laplace transforms of the value of the dynamic fund protection are obtained through martingale technique. Moreover, we analyze the value of dynamic fund protections under a generalized regime-switching jump diffusion model. Due to the complexity of Markov regime-switching, the jump process involved, and the nonlinearity, closed-form formulas for dynamic fund protection prices are virtually impossible to obtain. We design a numerical algorithm according to the Markov chain approximation techniques and obtain numerical results of the value of dynamic fund protection.

A Regime-Switching Diffusion Process Model for Advection-Dispersion Phenomena in Open Channels with Aquatic Vegetation

A Regime-Switching Diffusion Process Model for Advection-Dispersion Phenomena in Open Channels with Aquatic Vegetation

Transportation-cost inequalities for diffusions with jumps and its application to regime-switching processes

By using the Malliavin calculus, we establish the T1 transportation-cost inequalities for stochastic differential equations with jumps, which generalizes the results in [14]. As an application, we establish the T1 transportation-cost inequalities for regime-switching diffusion processes.

Criteria for transience and recurrence of regime-switching diffusion processes

We provide some on-off type criteria for recurrence of regime-switching diffusion processes using the theory of M-matrix, the Perron-Frobenius theorem. State-independent and state-dependent regime-switching diffusion processes in a finite space and an infinite countable space are both studied. Especially, we put forward a finite partition method to deal with switching process in an infinite countable space. As an application, we study the recurrence of regime-switching Ornstein-Uhlenbeck process, and provide an on-off type criterion for a kind of nonlinear regime-switching diffusion processes.