Regime-switching Diffusion Research Articles

Pricing double-barrier option with processes depending on various states of the economy

In this paper, we consider the pricing of double-barrier options under a Markov-modulated regime switching diffusion model. The proposed model incorporates structural changes in economic conditions and business and investment environments into the diffusion process, which captures some important stylized facts on asset returns such as asymmetry and heavy tail. Under the proposed model, we for the first time provide the closed-form approximation to double-barrier options, which can be also further improved systemically. The numerical results and the empirical example present that our pricing solution is highly effective and our work substantially extends the existed studies.

A General Stochastic Maximum Principle for Mean-Field Controls with Regime Switching

Focusing on regime-switching diffusions with mean-fields interactions, this paper is devoted to obtaining a general maximum principle. A main feature is that conditional mean-field is used in the controlled dynamic systems and the optimization process. Analysis of variational and adjoint equations using forward and backward stochastic differential equations with regime-switching and conditional mean-field is carried out. Necessary conditions for optimality are obtained without assuming the convexity of the control space. An example on conditional mean-variance portfolio selection with regime switching is given to illustrate the sufficient conditions for optimality.

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.

On Optimality of Threshold Dividend Strategy for Regime-Switching Diffusions With Negative Exponential Jumps

On Optimality of Threshold Dividend Strategy for Regime-Switching Diffusions With Negative Exponential Jumps

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.

Ergodicity of a Nonlinear Stochastic SIRS Epidemic Model with Regime-Switching Diffusions

In this paper, taking both white noises and colored noises into consideration, a nonlinear stochastic SIRS epidemic model with regime switching is explored. The threshold parameter R s is found, and we investigate sufficient conditions for the existence of the ergodic stationary distribution of the positive solution. Finally, some numerical simulations are also carried out to demonstrate the analytical results.

The Dynkin game with regime switching and applications to pricing game options

This paper is concerned with the Dynkin game (a zero-sum optimal stopping game). The dynamic of the system is modeled by a regime switching diffusion, in which the regime switching mechanism provides the structural changes of the random environment. The goal is to find a saddle point for the payoff functional up to one of the players exiting the game. Taking advantage of the method of penalization and the dynamic programming principle, the value function of the game problem is shown to be the unique viscosity solution to the associated variational inequalities. We also consider a financial example of pricing game option under a regime switching market. Both optimal stopping rules for the buyer and the seller and fair price of the option are numerically demonstrated in this example. All the results are markedly different from the traditional cases without regime switching.

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.

Integral equation characterization of the Feynman–Kac formula for a regime-switching diffusion

In this paper, we provide an integral equation characterization of the solution to a Cauchy problem associated to the Feynman–Kac formula for a regime-switching diffusion. We give a sufficient condition to guarantee the uniqueness of solutions to the integral equation and provide an example in the context of option pricing under the Ornstein–Uhlenbeck regime-switching model.

An MCMC computational approach for a continuous time state-dependent regime switching diffusion process

ABSTRACT State-dependent regime switching diffusion processes or hybrid switching diffusion (HSD) processes are hard to simulate with classical methods which leads us to adopt a Markov chain Monte Carlo (MCMC) Bayesian approach very convenient to estimate complicated models such as the HSD one. In the HSD, the diffusion component is dependent on the switching discrete hidden regimes and the transition rates of the regime switching are dependent on the diffusion observations. Since in reality phenomena are only observed in discrete times, data imputation is called for to create more observations so as to have good approximations for the density of the diffusion process. Three categories of entities will be computed in a Bayesian context: The latent imputed observations, the regime switching states, and the parameters of the models. The latent imputed data is updated at random time intervals in block using a Metropolis Hastings algorithm. The switching states are computed by an adaptation of a forward filtering backward smoothing algorithm to the HSD model. The parameters are estimated after prior specifications and conditional posterior densities formulation using Gibbs sampler or Metropolis Hastings algorithm.

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.

Stabilisation of coupled delayed regime‐switching diffusion with continuous‐state‐dependent switching via intermittent control

Motivated by Markovian regime-switching diffusion (MRSD), time-varying delays and non-linear coupling forms, this study is concerned with stabilisation of coupled regime-switching diffusion with time-varying delays (CRSDTD) under periodically intermittent control. Compared with most papers investigating MRSD, a striking feature of regime-switching diffusion is that in the two-component process ( x ( t ) , α ( t ) ) , switching process α ( t ) depends on the diffusion x ( t ) , which can be regarded as the x-dependent process briefly. It is worth mentioning that time-varying delays and non-linear coupling forms are taken into consideration when modelling, which makes their work more realistic. Moreover, a periodically intermittent controller is adopted to guarantee that CRSDTD can achieve the pth moment exponential stability and some sufficient conditions are derived by using the Lyapunov method and Kirchhoff's matrix tree theorem. Furthermore, to illustrate the applicability of their theoretical results, an application to regime-switching coupled oscillators with time-varying delays (RSCOTD) is presented and a stability criterion is given. Conclusively, numerical simulations on RSCOTD are presented to illustrate the effectiveness of the theoretical results.

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.

Stability of Regime-Switching Diffusion Systems with Discrete States Belonging to a Countable Set

This work focuses on the stability of regime-switching diffusions consisting of continuous and discrete components, in which the discrete component switches in a countably infinite set and its swit...

Optimal harvesting strategies for stochastic ecosystems

Focusing on multidimensional stochastic Lotka–Volterra type ecosystems in which multiple species are interacting, this study aims developing optimal harvesting strategies. The underlying systems are controlled regime-switching diffusions. Dealing with such a class of singular controls, the authors' objective is to maximise the total expected discounted rewards by using the ideas from singular control theory. In this study, the finiteness of the value function is obtained. The value function is characterised as a viscosity solution of the associated coupled systems of quasi-variational inequalities. Chattering policies are developed, which provide candidates for lower bounds of value functions. The impact of large noise on harvesting is studied by using certain upper bounds.

On the regularity of American options with regime-switching uncertainty

We study the regularity of the stochastic representation of the solution of a class of initial–boundary value problems related to a regime-switching diffusion. This representation is related to the value function of a finite-horizon optimal stopping problem such as the price of an American-style option in finance. We show continuity and smoothness of the value function using coupling and time-change techniques. As an application, we find the minimal payoff scenario for the holder of an American-style option in the presence of regime-switching uncertainty under the assumption that the transition rates are known to lie within level-dependent compact sets.