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In this paper, a periodic stochastic Lotka‐Volterra model with distributed delay and dispersal is investigated. First, by Itô's formula, we show that the solution with any positive initial value is global and positive. Then, we obtain the sufficient conditions guaranteeing the extinction of the system. Furthermore, we discuss the existence of the nontrivial periodic solution by Khasminskii theory. Finally, numerical examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

Robust analysis in the stochastic sense, including robust boundedness and robust stability, has become a momentous problem of stochastic systems. Up to now, almost all existing works about robust boundedness and robust stability require that stochastic perturbations are rooted in continuous time observations of systems states. However, compared with the continuous time noises case, stochastic perturbations of discrete time noises are not only appropriate but also reasonable. Hence, this brief proposes and explores the problems of robust boundedness and robust stability of discrete time noises for stochastic systems satisfying the linear growth condition. The destination of this brief is to answer the question: how much stochastic perturbations of discrete time noises can a bounded or stable system tolerate guaranteeing stochastically perturbed system remains asymptotically bounded or stable. In addition, this brief discusses the application of our results in neural networks. [ABSTRACT FROM AUTHOR]

Abstract After introducing an entropy formulation for stochastic nonlocal conservation laws via an integral formula, we develop a uniqueness theory by using a doubling variables technique and an existence theory by using a vanishing viscosity method. We further consider the continuous dependence of the entropy solution on initial data. [ABSTRACT FROM AUTHOR]

We introduce a notion of stochastic entropy solutions for heterogeneous scalar conservation laws with multiplicative noise on a bounded domain with non-homogeneous boundary condition. Using the concept of measure-valued solutions and Kruzhkov’s semi-entropy formulations, we show the existence and uniqueness of stochastic entropy solutions. Moreover, we establish an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux function and the random source function. [ABSTRACT FROM AUTHOR]

This work is about the existence of martingale solutions and weak solutions for a stochastic nonlocal Burgers equation on a one dimensional bounded domain. The existence of a martingale solution is shown by using a Galerkin approximation, Prokhorov's theorem and Skorokhod's embedding theorem. The same Galerkin approximation also leads to the existence of weak solution for the corresponding deterministic nonlocal Burgers equation on a bounded domain. [ABSTRACT FROM AUTHOR]

We discuss discrete stochastic processes with two independent variables: one is the standard symmetric random walk, and the other is the Poisson process. Convergence of discrete stochastic processes is analysed, such that the symmetric random walk tends to the standard Brownian motion. We show that a discrete analogue of Ito’s formula converges to the corresponding continuous formula. [ABSTRACT FROM AUTHOR]

This paper studies a stochastic two-species Monod competition chemostat model which is subject to environment noises. Such noises are described by independent standard Brownian motions. It proves that the initial value problem of the model has a unique positive global solution. However, unlike the corresponding deterministic model, the stochastic model no longer has positive equilibrium points. The asymptotic behaviors and the steady state distributions are established by using Itô’s formula, Lyaponov method and Gronwall inequality. In addition, numerical simulations are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we analyse a stochastic model driven by fractional Brownian motion representing HIV internal virus dynamics. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modeling. We show that the model established in this paper possesses non-negative solutions as this is essential in any population dynamics model. We also carry out analysis on the asymptotic behaviour of the model. We approximate one of the variables by a mean reverting process and find out the mean and variance of this process. [ABSTRACT FROM AUTHOR]

This paper is concerned with the renormalized stochastic entropy solutions of stochastic scalar conservation law forced by a multiplicative noise on a bounded domain with a non-homogeneous boundary condition. We first introduce a notion of renormalized stochastic entropy solutions and then establish the existence and uniqueness of the renormalized stochastic entropy solutions for a general L 1 -data. Our results allow us to give a positive answer to an open problem posed by Bauzet, Vallet and Wittbold in [4] . [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the Dirichlet boundary value problem for a multi-dimensional nonlocal conservation law with a multiplicative stochastic perturbation in a bounded domain. Using the concept of measure-valued solutions and Kruzhkov's semi-entropy formulations, a result of existence and uniqueness of entropy solution is proved. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a stochastic SIR epidemic model with saturated incidence is investigated. Firstly, we prove that the system has a unique global positive solution with any positive initial value. Then we verify that random effect may lead the disease to extinction under a simple condition. Thirdly, we establish a sufficient condition for persistence in the mean of the disease. Moreover, we show that there is a stationary distribution to the stochastic system under certain parametric restrictions. Finally, some numerical simulations are carried out to confirm the analytical results. [ABSTRACT FROM AUTHOR]

In this letter, a stochastic mutualism model with Lévy jumps and harvesting is considered. Under some simple assumptions, sufficient and necessary criteria for the existence of optimal harvesting policy are established. The optimal harvesting effort and the maximum of sustainable yield are also obtained. The effects of random noises on the optimal harvesting of the model are discussed and some numerical simulations are introduced to illustrate the main results. [ABSTRACT FROM AUTHOR]

We prove an existence and uniqueness result for quasilinear Stochastic PDEs with obstacle (in short OSPDE) under a weaker integrability condition on the coefficient and the barrier. [ABSTRACT FROM AUTHOR]

In this paper, we develop and study a stochastic model for the competition of three species with a generalized dose-response function in a polluted environment. We first carry out the survival analysis and obtain sufficient conditions for the extinction, non-persistence, weak persistence in the mean, strong persistence in the mean and stochastic permanence. The threshold between weak persistence in the mean and extinction is established for each species. Then, using Hasminskii's methods and a Lyapunov function, we derive sufficient conditions for the existence of stationary distribution for each population. Numerical simulations are carried out to support our theoretical results, and some biological significance is presented. [ABSTRACT FROM AUTHOR]

In this paper, the deterministic and stochastic SIRS epidemic models with nonlinear incidence are introduced and investigated. For deterministic system, the basic reproductive number R 0 is obtained. Furthermore, if R 0 ≤ 1 , then the disease-free equilibrium is globally asymptotically stable and if R 0 > 1 , then there is a unique endemic equilibrium which is globally asymptotically stable. For stochastic system, to begin with, we verify that there is a unique global positive solution starting from the positive initial value. Then when R 0 > 1 , we prove that stochastic perturbations may lead the disease to extinction in scenarios where the deterministic system is persistent. When R 0 ≤ 1 , a result on fluctuation of the solution around the disease-free equilibrium of deterministic model is obtained under appropriate conditions. At last, if the intensity of the white noise is sufficiently small and R 0 > 1 , then there is a unique stationary distribution to stochastic system. [ABSTRACT FROM AUTHOR]

This paper is concerned with effects of noise on the solutions of partial differential equations. We first provide a sufficient condition to ensure the existence of a unique positive solution for a class of stochastic partial differential equations. Then, we prove that noise could induce singularities (finite time blow up of solutions). Finally, we show that a stochastic Allen–Cahn equation does not have finite time singularities and the unique solution exists globally. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we focus on the stability problem for a class of stochastic delayed recurrent neural networks. Different from the traditional stability criteria, we introduce and study a new stability criterion: the mean-square exponential input-to-state stability. To the best of our knowledge, this new stability criterion has never been discussed in the field of stochastic recurrent neural networks. The main objective of the paper is to fill the gap. With the help of the Lyapunov–Krasovskii functional, stochastic analysis theory and Itô's formula, we prove that the addressed system is mean-square exponentially input-to-state stable. Moreover, two numerical examples and their simulations are presented to verify the theoretical results well. [Copyright &y& Elsevier]

Abstract: We study the stability of receding horizon control for continuous-time non-linear stochastic differential equations. We illustrate the results with a simulation example in which we employ receding horizon control to design an investment strategy to repay a debt. [Copyright &y& Elsevier]

In this paper, we present a differential infectivity SIR epidemic model with modified saturation incidences and stochastic perturbations. We show that the stochastic epidemic model has a unique global positive solution, and we utilize stochastic Lyapunov functions to show the asymptotic behavior of the solution. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a stochastic multi-group Lotka–Volterra mutualistic system under regime switching. It is well known that the population is forced to expire when the perturbation is sufficiently large. The main aim here is to investigate its ergodic property and positive recurrence by stochastic Lyapunov functions under small perturbation, which can be used to explain some recurring phenomena in practice and thus provide a good description of permanence. The mean of the stationary distribution is estimated. Simulations are also carried out to confirm our analytical results. [Copyright &y& Elsevier]

Abstract: In this paper, we obtain sufficient criteria for the existence of periodic solutions to deterministic SIR and SEIR epidemic models with modified saturation incidence rates by means of using the continuation theorem based on coincidence degree theory, and we show that the solution is unique and globally stable. Second, we discuss their corresponding stochastic epidemic models with random perturbation have a unique global positive solution respectively, and we utilize stochastic Lyapunov functions to investigate the asymptotic behavior of the solution. [Copyright &y& Elsevier]

This article discusses the exponential stability of nonlinear stochastic delay differential systems (SDDSs) whose coefficients obey the polynomial growth condition. Delay-dependent criteria on almost sure exponential stability and pth moment exponential stability of such SDDSs have been established. By applying some novel techniques, our criteria work for many SDDSs including some cases in which the ℒV operator has a complicated form, which seemingly prevents the existing results from being directly used. The range of order of moment exponential stability and the decay rate can be estimated through the coefficients of the system. [ABSTRACT FROM PUBLISHER]

Abstract: This paper addresses the existence and uniqueness of strong solutions to the stochastic variational inequality in a bounded domain with Dirichlet homogeneous conditions. Here for , , for . An application to the one-phase Stefan problem with a stochastic heat source is given. One studies also the corresponding stochastic parabolic equation with Signorini boundary conditions on . [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the problem of suppression explosive solutions by noise for nonlinear deterministic differential system. Given a deterministic differential system with coefficients satisfying a more general one-sided polynomial growth condition, we introduce Brownian noise feedback and therefore stochastically perturb this system into the nonlinear stochastic differential system . We show that appropriate guarantee that this stochastic system exists as a unique global solution although the corresponding deterministic systems may explode in a finite time. Under some weaker conditions, we reveal that the single noise can also make almost every path of the solution of corresponding stochastically perturbed system grow at most polynomially. [Copyright &y& Elsevier]

Abstract: The paper is concerned with the problem of non-existence of global solutions for a class of stochastic reaction–diffusion equations of Itô type. Under some sufficient conditions on the initial state, the nonlinear term and the multiplicative noise, it is proven that, in a bounded domain , there exist positive solutions whose mean -norm will blow up in finite time for , while, if , the previous result holds in any compact subset of . Two examples are given to illustrate some application of the theorems. [ABSTRACT FROM AUTHOR]

Itô's Formula is the stochastic analogue of the change of variable formula for deterministic integrals. It is a useful tool in dealing with stochastic integration. In this paper, using Henstock's approach, we derive two versions of Itô's Formula. Henstock's or generalized Riemann approach has been successful in giving an alternative definition of stochastic integral, which is more explicit, intuitive and less measure theoretic. Henstock's approach provides a simpler and more direct proof of Itô's Formula, although we do not claim that it is a generalization of the classical results. [ABSTRACT FROM AUTHOR]

Abstract: We study a crystalline version of the modified Stefan problem in the plane. The feature of our approach is that, we consider a model with stochastic perturbations and assume the interfacial curve to be a polygon. The existence of solution to our stochastic system is established. Galerkin’s method is one of the main tools used in the proof of our assertion. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation. We show there is a unique positive solution to the system with positive initial value, and mainly investigate the long time behavior of the system. Condition for the system to be extinct is given and persistent condition is established. At last, numerical simulations are carried out to support our results. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the stability analysis issue for stochastic delayed bidirectional associative memory (BAM) neural network with Markovian jumping parameters. Assume that the jumping parameters are generated from continue-time discrete-state homogeneous Markov process and the delays are time-invariant. By employing the Lyapunov stability theory, some inequality techniques and the stochastic analysis, sufficient conditions are derived to achieve the global exponential stability in the mean square of the stochastic BAM neural network. One example is also provided in the end of this paper to illustrate the effectiveness of our results. [Copyright &y& Elsevier]

In this article we stochastically perturb the classical non-autonomous delay Lotka-Volterra model [image omitted] into the stochastic delay population system (SDPS) [image omitted] Different from most of the existing papers [A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, J. Math. Anal. Appl. 292 (2004), 364-380, A. Bahar and X. Mao, Stochastic delay population dynamics, J. Pure Appl. Math. 11 (2004), 377-400, X. Mao, Delay population dynamics and environmental noise, Stochastics Dyn. 5(2) (2005), pp. 149-162], the system parameters in this article are time-dependent. We will give a sufficient condition under which the SDPS will have a unique global positive solution. We will then establish some new asymptotic properties for the moments of the solution. In particular, we will discuss two fundamental problems in population systems, namely ultimate boundedness and extinction. [ABSTRACT FROM AUTHOR]

Abstract: We develop a notion of nonlinear expectation–-expectation–generated by a nonlinear heat equation with infinitesimal generator . We first study multi-dimensional -normal distributions. With this nonlinear distribution we can introduce our -expectation under which the canonical process is a multi-dimensional -Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Itô’s type with respect to our -Brownian motion, and derive the related Itô’s formula. We have also obtained the existence and uniqueness of stochastic differential equations under our -expectation. [Copyright &y& Elsevier]

In this paper, we introduce a stochastic integral with respect to the solution X of the fractional heat equation on [0,1], interpreted as a divergence operator. This allows to use the techniques of the Malliavin calculus in order to establish an Ito-type formula for the process X. [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]

Abstract: This paper studies a class of stochastic delay differential equations with jumps (SDDEJs). Explicit solutions can hardly be obtained for the SDDEJs. Appropriate numerical approximation schemes such as the Euler scheme are needed to apply SDDEJs in practice or to study their properties. In this paper, it is proved that the Euler approximation solutions converge to the analytic solution for SDDEJs under weaker conditions than the linear growth condition and global Lipschitz condition. An example is given for illustration. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the linear stochastic heat equation with additive noise in dimension one. Then, using the representation of its solution as a stochastic convolution of the cylindrical Brownian motion with respect to an operator-valued kernel, we derive Itô''s- and Tanaka''s-type formulae associated to . [Copyright &y& Elsevier]

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]

The Itô formula for semimartingales is applied to develop equations for the characteristic function of the state of linear and non-linear dynamic systems with Gaussian, Poisson, and Lévy white noise, viewed as the formal derivatives of Brownian, compound Poisson, and Lévy processes, respectively. These equations can be obtained if the drift and diffusion coefficient of a dynamic system are polynomials of the system state and the driving noise is Gaussian or Poisson. It was not possible to derive equations for the characteristic function for the state of systems driven by Lévy white noise. Numerical results are presented for dynamic systems with real-valued states driven by Gaussian, Poisson, and Lévy white noise processes. [Copyright &y& Elsevier]