Stability of regime-switching diffusions

Khas’minskiĭ, R. Z.; Zhu, Chao; Yin, George

doi:10.1016/j.spa.2006.12.001

Cited by 188 publications

(85 citation statements)

References 8 publications

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“…Lemma 3 (see [78]). Suppose that the coefficients of (12) are -periodic in and there exists a function ( , ) ∈ 2 which is -periodic in , and ( , ) satisfies the following conditions:…”

Section: Lemma 2 Suppose Thatmentioning

confidence: 99%

“…Furthermore, the prey-predator model may be perturbed by telegraph noise which is distinguished by factors such as rain falls and nutrition and can be represented by switching among two or more regimes of environment [40,60,[70][71][72][73][74][75][76][77][78][79][80]. For example, population growth rates in different seasons are not the same.…”

Section: Introduction and Model Formulationmentioning

confidence: 99%

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Dynamical Analysis of a Class of Prey-Predator Model with Beddington-DeAngelis Functional Response, Stochastic Perturbation, and Impulsive Toxicant Input

et al. 2017

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A stochastic prey-predator system in a polluted environment with Beddington-DeAngelis functional response is proposed and analyzed. Firstly, for the system with white noise perturbation, by analyzing the limit system, the existence of boundary periodic solutions and positive periodic solutions is proved and the sufficient conditions for the existence of boundary periodic solutions and positive periodic solutions are derived. And then for the stochastic system, by introducing Markov regime switching, the sufficient conditions for extinction or persistence of such system are obtained. Furthermore, we proved that the system is ergodic and has a stationary distribution when the concentration of toxicant is a positive constant. Finally, two examples with numerical simulations are carried out in order to illustrate the theoretical results.

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“…Lemma 3 (see [78]). Suppose that the coefficients of (12) are -periodic in and there exists a function ( , ) ∈ 2 which is -periodic in , and ( , ) satisfies the following conditions:…”

Section: Lemma 2 Suppose Thatmentioning

confidence: 99%

Section: Introduction and Model Formulationmentioning

confidence: 99%

Dynamical Analysis of a Class of Prey-Predator Model with Beddington-DeAngelis Functional Response, Stochastic Perturbation, and Impulsive Toxicant Input

et al. 2017

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“…Therefore by virtue of Theorem 4.3 in Khasminskii et al (2007), the trivial solution of (4.21) is unstable in probability, which, in turn, implies that the trivial solution cannot be almost surely exponential stable. Therefore thanks to Proposition 4.2, the trivial solution of (4.21) is almost surely exponentially stable.…”

Section: Examplesmentioning

confidence: 99%

Almost Sure and Moment Exponential Stability of Regime-Switching Jump Diffusions

Chao¹,

Wang²,

Zhu³

et al. 2017

SIAM J. Control Optim.

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This work is devoted to almost sure and moment exponential stability of regimeswitching jump diffusions. The Lyapunov function method is used to derive sufficient conditions for stabilities for general nonlinear systems; which further helps to derive easily verifiable conditions for linear systems. For one-dimensional linear regime-switching jump diffusions, necessary and sufficient conditions for almost sure and pth moment exponential stabilities are presented. Several examples are provided for illustration.

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“…Ghosh et al 1993Ghosh et al , 1997. Theoretical properties of these dynamical systems have been recently studied in Khasminskii et al (2007). Our main result is a characterization of the Markov chain generator to get a target mixture marginal distribution for the diffusive component X(t).…”

Section: Introductionmentioning

confidence: 99%

Mixture Dynamics and Regime Switching Diffusions with Application to Option Pricing

Ramponi

2009

Methodol Comput Appl Probab

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In this paper we present a class of regime switching diffusion models described by a pair (X(t), Y(t)) ∈ R n × S, S = {1, 2, . . . , N}, Y(t) being a Markov chain, for which the marginal probability of the diffusive component X(t) is a given mixture. Our main motivation is to extend to a multivariate setting the class of mixture models proposed by Brigo and Mercurio in a series of papers. Furthermore, a simple algorithm is available for simulating paths through a thinning mechanism. The application to option pricing is considered by proposing a mixture version for the Margrabe Option formula and the Heston stochastic volatility formula for a plain vanilla.

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Stability of regime-switching diffusions

Cited by 188 publications

References 8 publications

Dynamical Analysis of a Class of Prey-Predator Model with Beddington-DeAngelis Functional Response, Stochastic Perturbation, and Impulsive Toxicant Input

Dynamical Analysis of a Class of Prey-Predator Model with Beddington-DeAngelis Functional Response, Stochastic Perturbation, and Impulsive Toxicant Input

Almost Sure and Moment Exponential Stability of Regime-Switching Jump Diffusions

Mixture Dynamics and Regime Switching Diffusions with Application to Option Pricing

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