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This work develops a generalization of the Goodwin–Lotka–Volterra models. The traditional model describes the evolution of the employment rate and the labor share. Motivated by recent developments on business cycles and Goodwin-type models, we consider a generalized Goodwin business cycle model with noise. We aim at investigating the impact of telegraph noise and white noise on the evolution of trajectories and the energy on trajectories of the model. It is proved that with the presence of the noises, all trajectories exit any compact subset of the open unit square D = ( 0 , 1 ) × ( 0 , 1 ) and the energy along trajectories can be arbitrarily large. Moreover, under certain additional conditions, the boundary of D is the stable limit cycle of the perturbed model by white noise. Noise impact on the period of business cycles is also discussed. Numerical examples are provided to demonstrate our findings. [ABSTRACT FROM AUTHOR]

Abstract: This work develops almost sure stability criteria for switching-jump-diffusion systems with state-dependent switching. By means of introducing certain auxiliary Markov chains and constructing certain order-preserving couplings, upper and lower “stability envelops” are constructed, which lead to systems with “upper and lower” approximating Markov chains. Using these approximations, sufficient conditions that are relatively easily verifiable for the almost sure stability and instability are obtained. When the jump process is missing, it is demonstrated that the techniques work equally well and provide a way to analyze the corresponding switching diffusion systems with -dependent switching. In addition, stochastic stabilization and destabilization are examined. Moreover, illustrative examples are provided for demonstration. [Copyright &y& Elsevier]

Abstract: Populations of biological species are often subject to different types of environmental noises. These noises play crucial roles and have significant impact on the evolution and biodiversity. To understand the effects of different types of noises on the asymptotic properties of the populations, this paper reveals the influence of inherent net birth noise and the interaction noise among stochastically perturbed population systems. By considering systems with infinite delays, (1) this paper discovers that (a) the interaction noise may suppress the potential explosion and guarantee the existence of the global solution, which yields regularity of the stochastic system, and (b) the inherent net birth noise may lead to extinction of the populations; (2) this paper provides sufficient conditions that ensure the stochastic persistence, stochastic boundedness, and asymptotic polynomial growth of the population system; (3) this paper demonstrates that these asymptotic properties are completely determined by stochastic noises and are independent of other parameters. [Copyright &y& Elsevier]

Abstract: This work examines almost sure stability of a pure random delay system whose delay time is modeled by a finite state continuous-time Markov chain with two-time scales. The Markov chain contains a fast-varying part and a slowly-changing part. Using the properties of the weighted occupation measure of the Markov chain, it is shown that the overall systemʼs almost-sure-asymptotic stability can be obtained by using the “averaged” delay. This feature implies that even if some longer delay times may destabilize the system individually, the system may still be stable if their impact is balanced. In other words, the Markov chain becomes a stabilizing factor. Numerical results are provided to demonstrate our results. [Copyright &y& Elsevier]

Abstract: This work is concerned with several properties of solutions of stochastic differential equations arising from hybrid switching diffusions. The word “hybrid” highlights the coexistence of continuous dynamics and discrete events. The underlying process has two components. One component describes the continuous dynamics, whereas the other is a switching process representing discrete events. One of the main features is the switching component depending on the continuous dynamics. In this paper, weak continuity is proved first. Then continuous and smooth dependence on initial data are demonstrated. In addition, it is shown that certain functions of the solutions verify a system of Kolmogorov''s backward differential equations. Moreover, rates of convergence of numerical approximation algorithms are dealt with. [Copyright &y& Elsevier]

Abstract: This work is concerned with stability of stochastic differential delay equations with Markovian switching, where the modulating Markov chain has a large state space and is subject to both fast and slow movements. Under simple conditions, we demonstrate that if the limit systems are pth-moment exponentially stable, then the original systems are pth-moment exponentially stable in an appropriate sense. In addition, the exponential stability is also investigated. Moreover, stability in distribution is obtained for such hybrid systems. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a class of nonlinear autoregressive (AR) processes with state-dependent switching, which are two-component Markov processes. The state-dependent switching model is a nontrivial generalization of Markovian switching formulation and it includes the Markovian switching as a special case. We prove the Feller and strong Feller continuity by means of introducing auxiliary processes and making use of the Radon–Nikodym derivatives. Then, we investigate the geometric ergodicity by the Foster–Lyapunov inequality. Moreover, we establish the -uniform ergodicity by means of introducing additional auxiliary processes and by virtue of constructing certain order-preserving couplings of the original as well as the auxiliary processes. In addition, illustrative examples are provided for demonstration. [Copyright &y& Elsevier]

Abstract: Focusing on competitive Lotka–Volterra model in random environments, this paper uses regime-switching diffusions to model the dynamics of the population sizes of n different species in an ecosystem subject to the random changes of the external environment. It is demonstrated that the growth rates of the population sizes of the species are bounded above. Moreover, certain long-run-average limits of the solution are examined from several angles. A partial stochastic principle of competitive exclusion is also derived. Finally, simple examples are used to demonstrate our findings. [Copyright &y& Elsevier]

Abstract: This work is concerned with diffusions with two-time scales or singularly perturbed diffusions. Asymptotic expansions of the solution of the associated Cauchy problem for parabolic partial differential equation are obtained and the desired error bounds are derived. These asymptotic expansions are then used to analyze related limit distributions of normalized integral functionals. [Copyright &y& Elsevier]

This paper develops exponential type upper bounds for scaled occupation measures of singularly perturbed Markov chains in discrete time. By considering two-time scale in the Markov chains, asymptotic analysis is carried out. The cases of the fast changing transition probability matrix is irreducible and that are divisible into l ergodic classes are examined first; the upper bounds of a sequence of scaled occupation measures are derived. Then extensions to Markov chains involving transient states and/or nonhomogeneous transition probabilities are dealt with. The results enable us to further our understanding of the underlying Markov chains and related dynamic systems, which is essential for solving many control and optimization problems. [Copyright &y& Elsevier]

This work is concerned with asymptotic properties of solutions to forward equations for singularly perturbed Markov chains with two small parameters. It is motivated by the model of a cost-minimizing firm involving production planning and capacity expansion and a two-level hierarchical decomposition. Our effort focuses on obtaining asymptotic expansions of the solutions to the forward equation. Different from previous work on singularly perturbed Markov chains, the inner expansion terms are constructed by solving certain partial differential equations. The methods of undetermined coefficients are used. The error bound is obtained. [Copyright &y& Elsevier]

This work focuses on population dynamics of two species described by Kolmogorov systems of competitive type under telegraph noise that is formulated as a continuous-time Markov chain with two states. Our main effort is on establishing the existence of an invariant (or a stationary) probability measure. In addition, the convergence in total variation of the instantaneous measure to the stationary measure is demonstrated under suitable conditions. Moreover, the Ω -limit set of a model in which each species is dominant in a state of the telegraph noise is examined in detail. [ABSTRACT FROM AUTHOR]