Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching

In this paper, we prove that a stochastic logistic population under regime switching controlled by a Markov chain is either stochastically permanent or extinctive, and we obtain the sufficient and necessary conditions for stochastic permanence and extinction under some assumptions. In the case of stochastic permanence we estimate the limit of the average in time of the sample path of the solution by two constants related to the stationary probability distribution of the Markov chain and the parameters of the subsystems of the population model. Finally, we illustrate our conclusions through two examples

Persistence, extinction and practical exponential stability of impulsive stochastic competition models with varying delays

This paper studies the persistence, extinction and practical exponential stability of impulsive stochastic competition models with time-varying delays. The existence of the global positive solutions is investigated by the relationship between the solutions of the original system and the equivalent system, and the sufficient conditions of system persistence and extinction are given. Moreover, our study shows the following facts: (1) The impulsive perturbation does not affect the practical exponential stability under the condition of bounded pulse intensity. (2) In solving the stability of non-Markovian processes, it can be transformed into solving the stability of Markovian processes by applying Razumikhin inequality. (3) In some cases, a non-Markovian process can produce Markovian effects. Finally, numerical simulations obtained the importance and validity of the theoretical results for the existence of practical exponential stability through the relationship between parameters, pulse intensity and noise intensity

Approximate solutions for a class of doubly perturbed stochastic differential equations

In this paper, we study the Carathéodory approximate solution for a class of doubly perturbed stochastic differential equations (DPSDEs). Based on the Carathéodory approximation procedure, we prove that DPSDEs have a unique solution and show that the Carathéodory approximate solution converges to the solution of DPSDEs under the global Lipschitz condition. Moreover, we extend the above results to the case of DPSDEs with non-Lipschitz coefficients

Razumikhin and Krasovskii stability of impulsive stochastic delay systems via uniformly stable function method

This paper generalizes Razumikhin-type theorem and Krasovskii stability theorem of impulsive stochastic delay systems. By proposing uniformly stable function (USF) in the form of impulse as a new tool, some properties about USF and some novel pth moment decay theorems are derived. Based on these new theorems, the stability theorems of impulsive stochastic linear delay system are acquired via the Razumikhin method and the Krasovskii method. The obtained results enhance the elasticity of the impulsive gain by comparing the previous results. Finally, numerical examples are given to demonstrate the effectiveness of theoretical results

Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation

In this paper, we are concerned with the asymptotic properties and numerical analysis of the solution to hybrid stochastic differential equations with jumps. Applying the theory of M-matrices, we will study the pth moment asymptotic boundedness and stability of the solution. Under the non-linear growth condition, we also show the convergence in probability of the Euler-Maruyama approximate solution to the true solution. Finally, some examples are provided to illustrate our new results

Some simple but challenging Markov processes

In this note, we present few examples of Piecewise Deterministic Markov Processes and their long time behavior. They share two important features: they are related to concrete models (in biology, networks, chemistry,. . .) and they are mathematically rich. Their math-ematical study relies on coupling method, spectral decomposition, PDE technics, functional inequalities. We also relate these simple examples to recent and open problems

Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state

Although the mean square stabilisation of hybrid systems by feedback controls based on discretetime observations of state and mode has been studied by several authors since 2013 (see, e.g., [17,19,27,31]), the corresponding almost sure stabilisation problem has little been investigated. Recent Mao [18] is the first to study the almost sure stabilisation of a given unstable system x(t) = f(x(t)) by a linear discretetime stochastic feedback control Ax([t/τ]τ)dB(t) (namely the stochastically controlled system has the form dx(t) = f(x(t))dt + Ax([t/τ]τ)dB(t)), where B(t) is a scalar Brownian, τ > 0 and [t/τ] is the integer part of t/τ. In this paper, we will consider a much more general problem. That is, we will to study the almost sure stabilisation of a given unstable hybrid system x(t) = f(x(t), r(t)) by nonlinear discrete-time stochastic feedback control u(x([t/τ]τ), r([t/τ]τ))dB(t) (so the stochastically controlled system is a hybrid stochastic system of the form dx(t) = f(x(t), r(t))dt + u(x([t/τ]τ), r([t/τ]τ))dB(t)), where B(t) is a multi-dimensional Brownian motion and r(t) is a Markov chain