A general autoregressive model with Markov switching: Estimation and consistency

In this paper, a general autoregressive model with Markov switching is considered, where the autoregression may be of an infinite order. The consistency of the maximum likelihood estimators for this model is obtained under regularity assumptions. Examples of finite and infinite order autoregressive models with Markov switching are discussed. Simulation studies with these examples illustrate the consistency and asymptotic normality of the estimators.      

Stochastic SIRS model under regime switching

We propose a new stochastic SIRS model under regime switching in which both white and color environmental noises are taken into account. We show that white noise suppresses explosions in the model and the disease-free equilibrium of the model is stochastically asymptotically stable under certain conditions. Moreover, we show that the model is stochastically ultimately bounded and the moment average in time of the model solution is bounded. An example illustrates the boundedness of the moment average in time of the model solution.     

Dynamics of a stochastic Lotka-Volterra model with regime switching

A stochastic Lotka-Volterra system under regime switching is proposed and investigated. First, sufficient conditions for stochastic permanence and extinction of the solution are established. Then the lower- and upper-growth rates of the positive solution are investigated. In addition, the superior limit of the average in time of the sample path of the solution is estimated. The results show that these properties have close relationships with the stationary probability distribution of the Markov chain. Finally, the main results are illustrated by several examples and figures. Some recent results are extended and improved.     

Mixture estimation with state-space components and Markov model of switching

The paper proposes a recursive algorithm for estimation of mixtures with state-space components and a dynamic model of switching. Bayesian methodology is adopted. The main features of the presented approach are: (i) recursiveness that enables a real-time performance of the algorithm; (ii) one-pass elaboration of the data sample; (iii) dynamic nature of the model of switching active components; (iv) orientation at explicit solutions with exploitation of numerical procedures only in those parts which cannot be computed analytically; (v) systematic approach to the Bayesian mixture estimation theory.     

Markowitz’s mean-variance asset-liability management with regime switching: A continuous-time model

This paper considers an asset-liability management (ALM) problem under a continuous-time Markov regime-switching model. By adopting the techniques of [Zhou, X.Y., Yin, G., 2003. Markowitz’s mean-variance portfolio selection with regime switching: A continuous-time model. SIAM J. Control Optim. 42, 1466–1482], we investigate the feasibility, obtain the optimal strategy, delineate the efficient frontier, and establish the associated mutual fund theorem.     

Markowitz’s mean-variance asset-liability management with regime switching: A continuous-time model

This paper considers an asset-liability management (ALM) problem under a continuous-time Markov regime-switching model. By adopting the techniques of [Zhou, X.Y., Yin, G., 2003. Markowitz’s mean-variance portfolio selection with regime switching: A continuous-time model. SIAM J. Control Optim. 42, 1466-1482], we investigate the feasibility, obtain the optimal strategy, delineate the efficient frontier, and establish the associated mutual fund theorem.     

Bayesian semiparametric Markov switching stochastic volatility model

This paper proposes a novel Bayesian semiparametric stochastic volatility model with Markov switching regimes for modeling the dynamics of the financial returns. The distribution of the error term of the returns is modeled as an infinite mixture of Normals; meanwhile, the intercept of the volatility equation is allowed to switch between two regimes. The proposed model is estimated using a novel sequential Monte Carlo method called particle learning that is especially well suited for state‐space models. The model is tested on simulated data and, using real financial times series, compared to a model without the Markov switching regimes. The results show that including a Markov switching specification provides higher predictive power for the entire distribution, as well as in the tails of the distribution. Finally, the estimate of the persistence parameter decreases significantly, a finding consistent with previous empirical studies.     

An interacting multiple model algorithm with a switching Markov chain

A Markov chain plays an important role in an interacting multiple model (IMM) algorithm which has been shown to be effective for target tracking systems. Such systems are described by a mixing of continuous states and discrete modes. The switching between system modes is governed by a Markov chain. In real world applications, this Markov chain may change or needs to be changed. Therefore, one may be concerned about a target tracking algorithm with the switching of a Markov chain. This paper concentrates on fault-tolerant algorithm design and algorithm analysis of IMM estimation with the switching of a Markov chain. Monte Carlo simulations are carried out and several conclusions are given.     

Analysis of a stochastic tumor-immune model with regime switching and impulsive perturbations

In this article, an impulsive stochastic tumor-immune model with regime switching is formulated and explored. Firstly, it is proven that the model has a unique global positive solution. Then sufficient criteria for extinction, non-persistence in the mean, weak persistence and stochastic permanence are provided. The threshold value between extinction and weak persistence is gained. In addition, the lower- and the upper-growth rates of tumor cells are estimated. The results demonstrate that the dynamics of the model are intimately associated with the random perturbations and impulsive perturbations. Finally, biological implications of the results are addressed with the help of real data and numerical simulations.     

VIX期权的状态转换随机波动率定价模型

基于Heston随机波动率模型提出了一种新的VIX期权定价模型,其中模型参数跟宏观经济状态有关,其状态方程满足连续时间的Markov Chain过程,在此基础上,得到了VIX看涨期权的定价公式.与传统的随机波动率模型相比,提出的期权定价公式中考虑了经济状态变换的风险溢价.最后,做了Monte Carlo数值模拟,并对数值结果进行了比较和解释.     

Modeling dependence structure among European markets and among Asian-Pacific markets: a regime switching regular vine copula approach

This paper investigates the structure of dependence among twelve European markets and among twelve Asian-Pacific markets. The dynamic of the dependence structure is described by a two-state regime switching model. The dependence structure during a bull phase is modelled by the Gaussian copula, while dependence during a bear phase is modelled by the regular vine copula. We analyze the regular vine structure in the second regime precisely. We perform a simplification procedure using a likelihood-ratio test and discuss the substitution of general regular vines by canonical vines or drawable vines. The analysis confirms the two-state nature of financial markets in addition to asymmetric and heavy-tailed dependences. Additionally, the European market has proven to be more strongly connected than the Asian-Pacific market, and European dependences are deeper in terms of conditional dependences. The results can be used by international investors by taking into account differences of both analyzed regions. Additionally, the analysis may help with the crisis prediction. The shift time to the market phase describing crisis times occurs significantly before the crisis itself.     

An integration by parts formula in a Markovian regime switching model and application to sensitivity analysis

We establish an integration by parts formula for the random functionals of a continuous-time Markov chain, based on partial differentiation with respect to jump times. In comparison with existing methods, our approach does not rely on the Girsanov theorem and it imposes less restrictions on the choice of directions of differentiation, while assuming additional continuity conditions on the considered functionals. As an application we compute sensitivities (Greeks) using stochastic weights in an asset price model with Markovian regime switching.     

Pricing swing options with regime switching

Gasoline price is highly volatile and exhibits Markov regime-switching process. In the electricity and the natural gas markets, “swing” options, which can provide some protection against day-to-day price fluctuations, are used to incorporate flexibility in delivering acquired energy. We propose a framework for pricing swing options for an underlying variable that follows a regime-switching process. We study the proposed framework in the gasoline industry for pricing swing options under price uncertainty by extracting the gasoline market information, estimating the parameters of the regime-switching process, and then presenting different numerical examples.     

Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching

This paper addresses a stochastic SIS epidemic model with vaccination under regime switching. The stochastic model in this paper includes white and color noises. By constructing stochastic Lyapunov functions with regime switching, we establish sufficient conditions for the existence of a unique ergodic stationary distribution.     

Optimal stock liquidation in a regime switching model with finite time horizon

This paper is concerned with a finite-horizon optimal selling rule. A set of geometric Brownian motions coupled by a finite-state Markov chain is used to characterize stock price movements. Given a fixed transaction fee, the optimal selling rule can be obtained by solving an optimal stopping problem. The corresponding value function is shown to be the unique viscosity solution to the associated HJB equations. Numerical solutions to these equations and their convergence are obtained. A numerical example is presented to illustrate the results.     

The threshold between permanence and extinction for a stochastic Logistic model with regime switching

A stochastic logistic model under regime switching is proposed and investigated. Sufficient conditions for extinction, non-persistence in the mean, weak persistence and stochastic permanence are established. The threshold between weak persistence and extinction is obtained. Then we show that this threshold also is the threshold between stochastic permanence and extinction under a simple additional condition. The results show that firstly, the stationary probability distribution of the Markov chain plays a key role in determining the permanence and extinction of the population. Secondly, different types of environmental noises have different effects on the permanence and extinction of the population. Thirdly, the more the stochastic noises, the easier the population goes to extinction.     

Numerical solutions for jump-diffusions with regime switching

This paper is devoted to numerical solutions for a class of jump-diffusions with regime switching. After briefly reviewing the notion of jump-diffusions with regime switching, finite-difference procedures are constructed. Under simple conditions, it is proved that the algorithm converges to the desired limit by means of a martingale problem formulation. Numerical experiments are carried out to demonstrate the performance of the algorithm.     

Ergodic stationary distribution and practical application of a hybrid stochastic cholera transmission model with waning vaccine-induced immunity under nonlinear regime switching

Considering the effect of stochasticity including white noise and colored noise, this paper aims to study a hybrid stochastic cholera epidemic model with waning vaccine-induced immunity and nonlinear telegraph perturbations. First, we derive a critical value ${?}_0^C$ related to the basic reproduction number ${?}_0$ of the deterministic model. The key aim of this paper is to generalize the θ-stochastic criterion method proposed by the recent work (Han et al. in Chaos Solit Fract 140:110238, 2020) to eliminate nonlinear telegraph perturbations. Next, via constructing several θ-stochastic Lyapunov functions and using the generalized method, we further prove that the stochastic model have a unique ergodic stationary distribution under ${?}_0^C>1$. Results show that the prevention and control of cholera epidemic depend on low transmission rate and small telegraph perturbations. Finally, the corresponding numerical simulations are performed to illustrate our analytical results and a practical application on the Somalia cholera outbreak is shown at the end of this paper.     

Ultracontractivity for Brownian motion with Markov switching

Consider the transition density functions for Brownian motion with two-state Markov switching. The characteristic functions for transition density functions are presented. Then, we show that the semigroup-associated Brownian motion with Markov switching is ultracontractive. And an explicit time-dependent upper bound for heat kernels are presented. Moreover, we prove that the Dirichlet form associated Brownian motion with Markov switching satisfies the Nash inequality.