In this paper,we consider the price of catastrophe options with credit risk in a regime-switching model.We assume that the macroeconomic states are described by a continuous-time Markov chain with a finite state space...In this paper,we consider the price of catastrophe options with credit risk in a regime-switching model.We assume that the macroeconomic states are described by a continuous-time Markov chain with a finite state space.By using the measure change technique,we derive the price expressions of catastrophe put options.Moreover,we conduct some numerical analysis to demonstrate how the parameters of the model affect the price of the catastrophe put option.展开更多
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 instabil...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.展开更多
This paper introduces a Bayesian Markov regime-switching model that allows the cointegration relationship between two time series to be switched on and off over time. Unlike classical approaches for testing and modeli...This paper introduces a Bayesian Markov regime-switching model that allows the cointegration relationship between two time series to be switched on and off over time. Unlike classical approaches for testing and modeling cointegration, the Bayesian Markov switching method allows for estimation of the regime-specific model parameters via Markov Chain Monte Carlo and generates more reliable estimation. Inference of regime switching also provides important information for further analysis and decision making.展开更多
In this paper, we establish properties for the switch-when-safe mean-variance strategies in the context of a Black-Scholes market model with stochastic volatility processes driven by a continuous-time Markov chain wit...In this paper, we establish properties for the switch-when-safe mean-variance strategies in the context of a Black-Scholes market model with stochastic volatility processes driven by a continuous-time Markov chain with a finite number of states. More precisely, expressions for the goal-achieving probabilities of the terminal wealth are obtained and numerical comparisons of lower bounds for these probabilities are shown for various market parameters. We conclude with asymptotic results when the Markovian changes in the volatility parameters appear with either higher or lower frequencies.展开更多
In this paper,we construct and analyze a Crank-Nicolson fitted finite volume scheme for pricing European options under regime-switching Kou’s jumpdiffusion model which is governed by a system of partial integro-diffe...In this paper,we construct and analyze a Crank-Nicolson fitted finite volume scheme for pricing European options under regime-switching Kou’s jumpdiffusion model which is governed by a system of partial integro-differential equations(PIDEs).We show that this scheme is consistent,stable and monotone as the mesh sizes in space and time approach zero,hence it ensures the convergence to the solution of continuous problem.Finally,numerical experiments are performed to demonstrate the efficiency,accuracy and robustness of the proposed method.展开更多
In this paper, the surplus of an insurance company is modeled by a Markovian regime- switching diffusion process. The insurer decides the proportional reinsurance and investment so as to increase revenue. The regime-s...In this paper, the surplus of an insurance company is modeled by a Markovian regime- switching diffusion process. The insurer decides the proportional reinsurance and investment so as to increase revenue. The regime-switching economy consists of a fixed interest security and several risky shares. The optimal proportional reinsurance and investment strategies with no short-selling constraints for maximizing an exponential utility on terminal wealth are obtained.展开更多
We study the pricing of an option when the price dynamic of the underlying risky asset is governed by a Markov-modulated geometric Brownian motion. We suppose that the drift and volatility of the underlying risky asse...We study the pricing of an option when the price dynamic of the underlying risky asset is governed by a Markov-modulated geometric Brownian motion. We suppose that the drift and volatility of the underlying risky asset are modulated by an observable continuous-time, finite-state Markov chain. We develop a two- stage pricing model which can price both the diffusion risk and the regime-switching risk based on the Esscher transform and the minimization of the maximum entropy between an equivalent martingale measure and the real-world probability measure over different states. Numerical experiments are conducted and their results reveal that the impact of pricing regime-switching risk on the option prices is significant.展开更多
This paper considers a proportional reinsurance-investment problem and an excess-of-loss reinsurance-investment problem for an insurer,where price processes of the risky assets and wealth process of the insurer are bo...This paper considers a proportional reinsurance-investment problem and an excess-of-loss reinsurance-investment problem for an insurer,where price processes of the risky assets and wealth process of the insurer are both described by Markovian regime switching.The target of the insurer is assumed to maximize the expected exponential utility from her terminal wealth with a state-dependent utility function.By employing the dynamic programming approach,the optimal value functions and the optimal reinsurance-investment strategies are derived.In addition,the impact of some parameters on the optimal strategies and the optimal value functions is analyzed,and lots of interesting results are discovered,such as the conclusion that excess-of-loss reinsurance is better than proportional reinsurance is not held in the regime-switching jump-diffusion model.展开更多
In this work,for a one-dimensional regime-switching diffusion process,we show that when it is positive recurrent,then there exists a stationary distribution,and when it is null recurrent,then there exists an invariant...In this work,for a one-dimensional regime-switching diffusion process,we show that when it is positive recurrent,then there exists a stationary distribution,and when it is null recurrent,then there exists an invariant measure. We also provide the explicit representation of the stationary distribution and invariant measure based on the hitting times of the process.展开更多
This paper extends the model and analysis of Lin, Tan and Yang （2009）. We assume that the financial market follows a regime-switching jump-diffusion model and the mortality satisfies Levy process. We price the point...This paper extends the model and analysis of Lin, Tan and Yang （2009）. We assume that the financial market follows a regime-switching jump-diffusion model and the mortality satisfies Levy process. We price the point to point and annual reset EIAs by Esscher transform method under Merton＇s assumption and obtain the closed form pricing formulas. Under two cases： with mortality risk and without mortality risk, the effects of the model parameters on the EIAs pricing are illustrated through numerical experiments.展开更多
This work is concerned with the continuous dependence on initial values of solutions of stochastic functional differential equations(SFDEs) with state-dependent regime-switching. Due to the state-dependence, this prob...This work is concerned with the continuous dependence on initial values of solutions of stochastic functional differential equations(SFDEs) with state-dependent regime-switching. Due to the state-dependence, this problem is very different to the corresponding problem for SFDEs without switching or SFDEs with Markovian switching. We provide a method to overcome the intensive interaction between the continuous component and the discrete component based on a subtle application of Skorokhod’s representation for jumping processes. Furthermore, we establish the strong convergence of Euler–Maruyama’s approximations, and estimate the order of error. The continuous dependence on initial values of Euler–Maruyama’s approximations is also investigated in the end.展开更多
We consider a two-dimensional reduced form contagion model with regime-switching interacting default intensities. The model assumes the intensities of the default times are driven by macro-economy described by a homog...We consider a two-dimensional reduced form contagion model with regime-switching interacting default intensities. The model assumes the intensities of the default times are driven by macro-economy described by a homogeneous Markov chain as well as the other default. By using the idea of ＇change of measure＇ and some closed-form formulas for the Laplace transforms of the integrated intensity processes, we derive the two-dimensional conditional and unconditional joint distributions of the default times. Based on these results, we give the explicit formulas for the fair spreads of the first-to-default and second-to-default credit default swaps （CDSs） on two underlyings.展开更多
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 ...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.展开更多
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 suitabl...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.展开更多
We investigate a particle system with mean field interaction living in a random environment characterized by a regime-switching process.The switching process is allowed to be dependent on the particle system.The well-...We investigate a particle system with mean field interaction living in a random environment characterized by a regime-switching process.The switching process is allowed to be dependent on the particle system.The well-posedness and various properties of the limit conditional McKean-Vlasov SDEs are studied,and the conditional propagation of chaos is established with explicit estimate of the convergence rate.展开更多
This paper introduces and represents conditional coherent risk measures as essential suprema of conditional expectations over a convex set of probability measures and as distorted expectations given a concave distorti...This paper introduces and represents conditional coherent risk measures as essential suprema of conditional expectations over a convex set of probability measures and as distorted expectations given a concave distortion function.A model is then developed for the bid and ask prices of a European-type asset by a conic formulation.The price process is governed by a modified geometric Brownian motion whose drift and diffusion coefficients depend on a Markov chain.The bid and ask prices of a European-type asset are then characterized using conic quantization.展开更多
The contagion credit risk model is used to describe the contagion effect among different financial institutions. Under such a model, the default intensities are driven not only by the common risk factors, but also by ...The contagion credit risk model is used to describe the contagion effect among different financial institutions. Under such a model, the default intensities are driven not only by the common risk factors, but also by the defaults of other considered firms. In this paper, we consider a two-dimensional credit risk model with contagion and regime-switching. We assume that the default intensity of one firm will jump when the other firm defaults and that the intensity is controlled by a Vasicek model with the coefficients allowed to switch in different regimes before the default of other firm. By changing measure, we derive the marginal distributions and the joint distribution for default times. We obtain some closed form results for pricing the fair spreads of the first and the second to default credit default swaps （CDSs）. Numerical results are presented to show the impacts of the model parameters on the fair spreads.展开更多
A regime-switching geometric Brownian motion is used to model a geometric Brownian motion with its coefficients changing randomly according to a Markov chain.In this work, the author gives a complete characterization ...A regime-switching geometric Brownian motion is used to model a geometric Brownian motion with its coefficients changing randomly according to a Markov chain.In this work, the author gives a complete characterization of the recurrent property of this process. The long time behavior of this process such as its p-th moment is also studied. Moreover, the quantitative properties of the regime-switching geometric Brownian motion with two-state switching are investigated to show the difference between geometric Brownian motion with switching and without switching. At last, some estimates of its first passage probability are established.展开更多
This paper is devoted to study the proportional reinsurance/new business and investment problem under the mean-variance criterion in a continuous-time setting.The strategies are constrained in the non-negative cone an...This paper is devoted to study the proportional reinsurance/new business and investment problem under the mean-variance criterion in a continuous-time setting.The strategies are constrained in the non-negative cone and all coefficients in the model except the interest rate are stochastic processes adapted the filtration generated by a Markov chain.With the help of a backward stochastic differential equation driven by the Markov chain,we obtain the optimal strategy and optimal cost explicitly under this non-Markovian regime-switching model.The cases with one risky asset and Markov regime-switching model are considered as special cases.展开更多
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 regimc...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 regimc-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 systerm without switching,the advantages of the stochastic regime-switching mutualism model are given.展开更多
基金supported by the Jiangsu University Philosophy and Social Science Research Project(Grant No.2019SJA1326).
文摘In this paper,we consider the price of catastrophe options with credit risk in a regime-switching model.We assume that the macroeconomic states are described by a continuous-time Markov chain with a finite state space.By using the measure change technique,we derive the price expressions of catastrophe put options.Moreover,we conduct some numerical analysis to demonstrate how the parameters of the model affect the price of the catastrophe put option.
基金the National Science Foundation (No. DMS-0603287, No. CMS-0510655)the National Security Agency (No. MSPF-068-029)+3 种基金the National Natural Science Foundation of China (No. 60574069)Program for NCET,in part by the Key Project of Chinese Ministry of Education 104053and in part by theWayne State University Research Enhancement Programthe National Science Foundation (No.DMS-0304928, No. DMS-0624849)
文摘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.
文摘This paper introduces a Bayesian Markov regime-switching model that allows the cointegration relationship between two time series to be switched on and off over time. Unlike classical approaches for testing and modeling cointegration, the Bayesian Markov switching method allows for estimation of the regime-specific model parameters via Markov Chain Monte Carlo and generates more reliable estimation. Inference of regime switching also provides important information for further analysis and decision making.
文摘In this paper, we establish properties for the switch-when-safe mean-variance strategies in the context of a Black-Scholes market model with stochastic volatility processes driven by a continuous-time Markov chain with a finite number of states. More precisely, expressions for the goal-achieving probabilities of the terminal wealth are obtained and numerical comparisons of lower bounds for these probabilities are shown for various market parameters. We conclude with asymptotic results when the Markovian changes in the volatility parameters appear with either higher or lower frequencies.
基金supported by the National Natural Science Foundation of China(Nos.11971354,and 11701221)the Special Basic Cooperative Research Programs of Yunnan Provincial Undergraduate Universities’Association(No.2019FH001-079)the Fundamental Research Funds for the Central Universities(No.22120210555).
文摘In this paper,we construct and analyze a Crank-Nicolson fitted finite volume scheme for pricing European options under regime-switching Kou’s jumpdiffusion model which is governed by a system of partial integro-differential equations(PIDEs).We show that this scheme is consistent,stable and monotone as the mesh sizes in space and time approach zero,hence it ensures the convergence to the solution of continuous problem.Finally,numerical experiments are performed to demonstrate the efficiency,accuracy and robustness of the proposed method.
基金supported by National Natural Science Foundation of China (Grant No.11001139)Fundamental Research Funds for the Central Universities (Grant No.65010771)+1 种基金Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP Grant No.20100031120002)the second author is supported by the Discovery Grant from the Australian Research Council (ARC) (Project No.DP1096243)
文摘In this paper, the surplus of an insurance company is modeled by a Markovian regime- switching diffusion process. The insurer decides the proportional reinsurance and investment so as to increase revenue. The regime-switching economy consists of a fixed interest security and several risky shares. The optimal proportional reinsurance and investment strategies with no short-selling constraints for maximizing an exponential utility on terminal wealth are obtained.
基金the Research Grants Councilof the Hong Kong Special Administrative Region,China(Project No.HKU 754008H)
文摘We study the pricing of an option when the price dynamic of the underlying risky asset is governed by a Markov-modulated geometric Brownian motion. We suppose that the drift and volatility of the underlying risky asset are modulated by an observable continuous-time, finite-state Markov chain. We develop a two- stage pricing model which can price both the diffusion risk and the regime-switching risk based on the Esscher transform and the minimization of the maximum entropy between an equivalent martingale measure and the real-world probability measure over different states. Numerical experiments are conducted and their results reveal that the impact of pricing regime-switching risk on the option prices is significant.
基金supported by the National Natural Science Foundation of China under Grant Nos.71501050 and 71231008the National Science Foundation of Guangdong Province of China under Grant No.2014A030310195+1 种基金Guangdong Natural Science for Research Team under Grant No.2014A030312003Chinese Scholarship Council under Grant No.201508440324
文摘This paper considers a proportional reinsurance-investment problem and an excess-of-loss reinsurance-investment problem for an insurer,where price processes of the risky assets and wealth process of the insurer are both described by Markovian regime switching.The target of the insurer is assumed to maximize the expected exponential utility from her terminal wealth with a state-dependent utility function.By employing the dynamic programming approach,the optimal value functions and the optimal reinsurance-investment strategies are derived.In addition,the impact of some parameters on the optimal strategies and the optimal value functions is analyzed,and lots of interesting results are discovered,such as the conclusion that excess-of-loss reinsurance is better than proportional reinsurance is not held in the regime-switching jump-diffusion model.
基金supported by National Natural Science Foundation of China(Grant No.11301030)Beijing Higher Education Young Elite Teacher Project
文摘In this work,for a one-dimensional regime-switching diffusion process,we show that when it is positive recurrent,then there exists a stationary distribution,and when it is null recurrent,then there exists an invariant measure. We also provide the explicit representation of the stationary distribution and invariant measure based on the hitting times of the process.
基金supported by National Natural Science Foundation of China (Grant Nos.10971068 and 11231005)Shanghai Municipal Natural Science Foundation (Grant No. 12ZR1408300)+3 种基金Humanity and Social Science Youth Foundation of Ministry of Education of China (Grant No. 12YJC910006)Doctoral Program Foundation of the Ministry of Education of China (Grant No. 20110076110004)Program for New Century Excellent Talents in University (Grant No. NCET-09-0356)the Fundamental Research Funds for the Central Universities
文摘This paper extends the model and analysis of Lin, Tan and Yang （2009）. We assume that the financial market follows a regime-switching jump-diffusion model and the mortality satisfies Levy process. We price the point to point and annual reset EIAs by Esscher transform method under Merton＇s assumption and obtain the closed form pricing formulas. Under two cases： with mortality risk and without mortality risk, the effects of the model parameters on the EIAs pricing are illustrated through numerical experiments.
基金Supported in part by NNSFs of China(Grant Nos.11771327,11431014,11831014)。
文摘This work is concerned with the continuous dependence on initial values of solutions of stochastic functional differential equations(SFDEs) with state-dependent regime-switching. Due to the state-dependence, this problem is very different to the corresponding problem for SFDEs without switching or SFDEs with Markovian switching. We provide a method to overcome the intensive interaction between the continuous component and the discrete component based on a subtle application of Skorokhod’s representation for jumping processes. Furthermore, we establish the strong convergence of Euler–Maruyama’s approximations, and estimate the order of error. The continuous dependence on initial values of Euler–Maruyama’s approximations is also investigated in the end.
基金Acknowledgements The authors thank the anonymous referees for valuable comments to improve the earlier version of the paper. The research of Yinghui Dong was supported by the Natural Science Foundation of Jiangsu Province （Grant No. BK20130260）, the National Natural Science Foundation of China （Grant No. 11301369）, and the China Postdoctoral Science Foundation （Grant No. 2013M540371）. The research of Guojing Wang was supported by the National Natural Science Foundation of China （Grant No. 11371274） and the Natural Science Foundation of Jiangsu Province （Grant No. BK2012613）.
文摘We consider a two-dimensional reduced form contagion model with regime-switching interacting default intensities. The model assumes the intensities of the default times are driven by macro-economy described by a homogeneous Markov chain as well as the other default. By using the idea of ＇change of measure＇ and some closed-form formulas for the Laplace transforms of the integrated intensity processes, we derive the two-dimensional conditional and unconditional joint distributions of the default times. Based on these results, we give the explicit formulas for the fair spreads of the first-to-default and second-to-default credit default swaps （CDSs） on two underlyings.
基金supported by National Natural Science Foundation of China (Grant Nos. 11301030, 11401169 and 11431014)Key Scientific Research Projects of Henan Province (Grant No. 16A110010)
文摘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.
基金This work was supported in part by the National Natural Science Foundation of China(Grant No.12071031).
文摘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.
基金supported in part by the National Natural Science Foundation of China(Grant Nos.11771327,11831014).
文摘We investigate a particle system with mean field interaction living in a random environment characterized by a regime-switching process.The switching process is allowed to be dependent on the particle system.The well-posedness and various properties of the limit conditional McKean-Vlasov SDEs are studied,and the conditional propagation of chaos is established with explicit estimate of the convergence rate.
文摘This paper introduces and represents conditional coherent risk measures as essential suprema of conditional expectations over a convex set of probability measures and as distorted expectations given a concave distortion function.A model is then developed for the bid and ask prices of a European-type asset by a conic formulation.The price process is governed by a modified geometric Brownian motion whose drift and diffusion coefficients depend on a Markov chain.The bid and ask prices of a European-type asset are then characterized using conic quantization.
基金Acknowledgements The authors cordially thank the anonymous reviewers for valuable comments to improve the earlier version of the paper. This work was supported by the National Natural Science Foundation of China （Grant Nos. 11371274, 11671291）, the Natural Science Foundation of Jiangsu Province （Grant No. BK20160300）, and the Open Project of Jiangsu Key Laboratory of Financial Engineering （Grant No. NSK2015-05）.
文摘The contagion credit risk model is used to describe the contagion effect among different financial institutions. Under such a model, the default intensities are driven not only by the common risk factors, but also by the defaults of other considered firms. In this paper, we consider a two-dimensional credit risk model with contagion and regime-switching. We assume that the default intensity of one firm will jump when the other firm defaults and that the intensity is controlled by a Vasicek model with the coefficients allowed to switch in different regimes before the default of other firm. By changing measure, we derive the marginal distributions and the joint distribution for default times. We obtain some closed form results for pricing the fair spreads of the first and the second to default credit default swaps （CDSs）. Numerical results are presented to show the impacts of the model parameters on the fair spreads.
基金supported by the National Natural Science Foundation of China(Nos.11301030,11431014)
文摘A regime-switching geometric Brownian motion is used to model a geometric Brownian motion with its coefficients changing randomly according to a Markov chain.In this work, the author gives a complete characterization of the recurrent property of this process. The long time behavior of this process such as its p-th moment is also studied. Moreover, the quantitative properties of the regime-switching geometric Brownian motion with two-state switching are investigated to show the difference between geometric Brownian motion with switching and without switching. At last, some estimates of its first passage probability are established.
基金supported by the 111 Project[grant number B14019]the National Natural Science Foundation of China[grant numbers 11571113,11601157,11601320].
文摘This paper is devoted to study the proportional reinsurance/new business and investment problem under the mean-variance criterion in a continuous-time setting.The strategies are constrained in the non-negative cone and all coefficients in the model except the interest rate are stochastic processes adapted the filtration generated by a Markov chain.With the help of a backward stochastic differential equation driven by the Markov chain,we obtain the optimal strategy and optimal cost explicitly under this non-Markovian regime-switching model.The cases with one risky asset and Markov regime-switching model are considered as special cases.
基金The work was supported in part by NSFC of China Grants 11771123Science and Technology Research Key Projects of the Education Department of Henan Province 2015GGJS-024+1 种基金the Project Funded by Cliina Postdoctoral Science Foundation Nos.2016M600427 and 2017T100385Postdoctoral Science Foundation of Jiangsu Province(Grant No.1601141B).
文摘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 regimc-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 systerm without switching,the advantages of the stochastic regime-switching mutualism model are given.
