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The aim of this article is to consider the existence, uniqueness and uniformly exponential p-stability of the solution for ∇ -delay stochastic dynamic equations on time scales via Lyapunov functions. This work can be considered as a unification and generalization of stochastic difference and stochastic differential time-varying delay equations. [ABSTRACT FROM AUTHOR]

This article studies the asymptotic behavior of a stochastic Chemostat model with Lotka–Volterra food chain in which the dilution rate was influenced by white noise. The long-time behavior of the model is studied. Using Lyapunov function and Itô's formula, we show that there is a unique positive solution to the system. Moreover, the sufficient conditions for some population dynamical properties including the boundedness in mean and the stochastically asymptotic stability of the washout equilibrium were obtained. Furthermore, we show how the solutions spiral around the predator-free equilibrium and the positive equilibrium of deterministic system. Besides, the existence of the stationary distribution is proved for the considered model. Numerical simulations are introduced finally to support the obtained results. [ABSTRACT FROM AUTHOR]

The main aim of this article is to deal with the almost-sure stability of stochastic differential delay equations. Our improved theorems give better results while conditions imposed on the Lyapunov function are much weaker, thus, it is easier to find a right Lyapunov function in application. [ABSTRACT FROM PUBLISHER]

The aim of this article is to study the Doob–Meyer decomposition theorem, ∇-stochastic integration and Ito's formula for stochastic processes defined on time scale. The obtained results can be considered as a first attempt on the stochastic calculus on time scale. [ABSTRACT FROM PUBLISHER]

In this article, we stochastically perturb the classical non-autonomous Lotka-Volterra model [image omitted] into the stochastic differential equation [image omitted] Different from most of existing articles, for example, [3, 20] the system parameters in this article are time-dependent. We will give a sufficient condition under which the stochastic differential equation will have a unique global positive solution. We will then establish some new asymptotic properties for the moments as well as for the sample paths of the solution. In particular, we will discuss two fundamental problems in population systems, namely ultimate boundedness and extinction. [ABSTRACT FROM AUTHOR]

In this paper, we will establish new results on the attraction for solutions to stochastic functional differential equations with respect to semimartingale. Most of the existing results stochastic stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in the study of attraction. Moreover, from our results on the attraction follow several new criteria on almost surely asymptotic stability and boundedness of the solutions. [ABSTRACT FROM AUTHOR]

The classical Khasminskii theorem (see [6]) on the nonexplosion solutions of stochastic differential equations (SDEs) is very important since it gives a powerful test for SDEs to have nonexplosion solutions without the linear growth condition. Recently, Mao [13] established a Khasminskii-type test for stochastic differential delay equations (SDDEs). However, the Mao test can not still be applied to many important SDDEs, e.g., the stochastic delay power logistic model in population dynamics. The main aim of this paper is to establish an even more general Khasminskii-type test for SDDEs that covers a wide class of highly nonlinear SDDEs. As an application, we discuss a stochastic delay Lotka-Volterra model of the food chain to which none of the existing results but our new Khasminskii-type test can be applied. [ABSTRACT FROM AUTHOR]

For a given stochastic process X, its segment Xt at time t, represents the "slice" of each path of X over a fixed time-interval [t - r, t], where r is the length of the "memory" of the process. Segment processes are important in the study of stochastic systems with memory (stochastic functional differential equations or SFDEs). The main objective of this paper is to study nonlinear transforms of segment processes. Toward this end, we construct a stochastic integral with respect to the Brownian segment process. The difficulty in this construction is the fact that the stochastic integrator is infinite dimensional and is not a (semi)martingale. We overcome this difficulty by employing Malliavin (anticipating) calculus techniques. The segment integral is interpreted as a Skorohod integral via a stochastic Fubini theorem. We then prove Itô's formula for the segment of a continuous Skorohod-type process and embed the segment calculus in the theory of anticipating calculus. Applications of the Ito formula include the weak infinitesimal generator for the solution segment of a stochastic system with memory, the associated Feynman-Kac formula, and the Black-Scholes PDE for stock dynamics with memory. [ABSTRACT FROM AUTHOR]