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Abstract In this paper, stochastic effect on the spread of the infectious disease with saturated incidence rate and the special transfer from infectious is discussed. The threshold dynamics is explored for the case of relatively small noise. Our results show that large noise will cause the elimination of the disease, which will help suppress the spread of the disease.

In this paper, we introduce the Itô-Henstock integral of an operator-valued stochastic process and formulate a version of Itô's formula.

Abstract In this paper, the deterministic and stochastic eco-epidemiological models with modified Leslie–Gower functional response are studied. For a deterministic system, the stability of disease-free equilibrium and positive equilibrium is studied. For a stochastic system, we verify that the system admits a unique positive global solution starting from any positive initial value, and we establish the conditions of extinction for infected prey population and strong persistence in mean for all species. We also show the system has a stationary distribution under some conditions. Finally, some numerical simulations are carried out to illustrate the main results.

In this paper, we solve the mean-square exponential input-to-state stability problem for a class of stochastic delayed recurrent neural networks with time-varying coefficients. With the aid of stochastic analysis theory and a Lyapunov-Krasovskii functional, we derive a novel criterion that ensures the given system is mean-square exponentially input-to-state stable. Furthermore, the new criterion generalizes and improves some known results. Finally, two examples and their numerical simulations are provided to demonstrate the theoretical results.

In this paper, we study the dynamics property of a stochastic HIV model with Beddington–DeAngelis functional response. It has a unique uninfected steady state. We prove that the model has a unique global positive solution. Furthermore, if the basic reproductive number is not larger than 1, the asymptotic behavior of the solution is stochastically stable. Otherwise, it fluctuates randomly around the infected steady state of the corresponding deterministic HIV model. Finally, some numerical simulations are carried out to verify our results. [ABSTRACT FROM AUTHOR]

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 paper, studies the class of diffusion processes generated by a first-order continuous-time bilinear processes (COBL(1, 1)) with time-varying coefficients. So, we used the It o ̂ formula approach for examining the L 2 − structure of the process and its powers. In time-invariant case, an expression of the moments of any order are given and the continuous autoregressive (C A R) representation of such version is given, in particular the moments properties of some specifications are however derived. Based on these results we are able to examine the statistical properties as well as we develop an estimation method of the process via the so-called Yule-Walker (YW) type algorithm which relates with unknown parameters of CAR representation. The method is illustrated by a Monte Carlo study and applied to modeling the electricity consumption sampled at each 15 mn in Algeria. [ABSTRACT FROM AUTHOR]

Let L = { L (t , x) , t ≥ 0 , x ∈ R } be the local time of a G-Brownian motion B on a sublinear expectation space (Ω , H , E ˆ). In this paper, we show that the local time L is a rough path of roughness p quasi-surely for any 2

In this paper, the effect of noise intensity on parabolic equations is considered. We focus on the effect of noise on the energy solutions of stochastic parabolic equations. By utilising Ito's formula and the energy estimate method, we obtain excitation indices of the solution u at time t. Furthermore, we improve existing results by introducing a simple method to verify the existing results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we use an Ornstein–Uhlenbeck process to describe the environmental stochasticity and propose a stochastic predator–prey model with modified Leslie–Gower and Holling-type II schemes. For each species, sharp sufficient conditions for persistence in the mean and extinction are respectively obtained. The results demonstrate that the persistence and extinction of the species have close relationships with the environmental stochasticity. In addition, the theoretical results are numerically illustrated by some simulations. [ABSTRACT FROM AUTHOR]

Guo et al. [8] are the first to study the strong convergence of the explicit numerical method for the highly nonlinear stochastic differential delay equations (SDDEs) under the generalised Khasminskii-type condition. The method used there is the truncated Euler–Maruyama (EM) method. In this paper we will point out that a main condition imposed in [8] is somehow restrictive in the sense that the condition could force the step size to be so small that the truncated EM method would be inapplicable. The key aim of this paper is then to establish the convergence rate without this restriction. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a version of Itô's formula for the backwards Itô-Henstock integral of an operator-valued stochastic process. Itô's formula is the stochastic analogue of the change of variable for deterministic integrals. [ABSTRACT FROM AUTHOR]

In this paper, we develop the theory of nonlinear rough paths. Following the ideas of Lyons and Gubinelli, we define the nonlinear rough integral ∫tsW(dr; Yr), where W and Y are only α-Hölder continuous in time with α ∈ (1/3, 1/2 ]. Also, we study the Kunita-type equation Yt =ξ + ∫0t W(dr, Ys), obtaining the local and global existence and uniqueness of the solution under suitable sufficient conditions. As an application, we study transport equations with rough vector fields and observe that the classical solution formula for smooth and Young's cases does not provide a solution to the rough equation. Indeed this formula satisfies a transport equation with additional compensator terms (see (1.7)). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the exponential stability of highly nonlinear hybrid neutral pantograph stochastic differential equations (NPSDEs). The aim of this paper is to establish exponential stability criteria for a class of hybrid NPSDEs without the linear growth condition. The methods of Lyapunov functions and M‐matrix are used to study exponential stability and boundedness of the hybrid NPSDEs. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of solution to obstacle problem for quasilinear stochastic partial differential equations with Neumann boundary condition. Our method is based on the analytical techniques coming from parabolic potential theory. The solution is expressed as a pair (u , ν) where u is a predictable continuous process which takes values in a proper Sobolev space and ν is a random regular measure satisfying minimal Skohorod condition. [ABSTRACT FROM AUTHOR]

In this paper, a two prey and one predator model with logistic growth in prey species have been considered in presence of stochastic perturbation. The cooperation among the preys against predation is assumed. Necessary conditions for asymptotic behavior around the equilibrium points of its deterministic system with and without stochasticity have been discussed. Finally, to illustrate the analytical results, numerical simulations have been performed. [ABSTRACT FROM AUTHOR]

Here, we review some results of fractional volatility models, where the volatility is driven by fractional Brownian motion (fBm). In these models, the future average volatility is not a process adapted to the underlying filtration, and fBm is not a semimartingale in general. So, we cannot use the classical Itô’s calculus to explain how the memory properties of fBm allow us to describe some empirical findings of the implied volatility surface through Hull and White type formulas. Thus, Malliavin calculus provides a natural approach to deal with the implied volatility without assuming any particular structure of the volatility. The aim of this paper is to provides the basic tools of Malliavin calculus for the study of fractional volatility models. That is, we explain how the long and short memory of fBm improves the description of the implied volatility. In particular, we consider in detail a model that combines the long and short memory properties of fBm as an example of the approach introduced in this paper. The theoretical results are tested with numerical experiments.

Abstract We propose a new stochastic competition chemostat system with saturated growth rate and impulsive toxicant input. The main purpose of this paper is to study the stochastic dynamics of a high-dimensional impulsive stochastic chemostat model and find the threshold between persistence and extinction for the impulsive stochastic chemostat system. First, we investigate the stability of the periodic solution of a deterministic impulsive chemostat model and obtain the threshold between persistence and extinction for the system. Second, by using qualitative analysis method of impulsive stochastic differential equations, we obtain conditions for the extinction and persistence in mean of two microorganisms in the stochastic chemostat model. The results show that a stochastic disturbance or the impulsive effect can cause the microorganisms to go to extinction. Finally, we provide some examples together with numerical simulations to illustrate the analytical results and explain the biological implications.

In this paper, we introduce a class of nonlinear population diffusion system. Existence and uniqueness of strong solution are proved for stochastic age-dependent population diffusion system in Hilbert space by use of Gronwall's lemma and Burkholder–Davis–Gundy's inequality.

In this paper, we aim to obtain the Hölder continuous of solutions to stochastic nonlocal heat equations. By using Campanato estimates and Sobolev embedding theorem, we first prove the Hölder continuous of the mild solution of stochastic nonlocal diffusion equations locally in R d in the sense that the solution u belongs to the space C γ (D T ; L p (Ω)). Then by using tail estimates, we obtain the estimates of the mild solution in L p (Ω ; C γ (D T)). Lastly, we prove the Hölder continuous of the mild solution on bounded domain. [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]

Abstract In this paper, we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises d u t (x) = Δ α 2 u t (x) d t + g (t , x) d η t , u 0 = 0 , t ∈ (0 , T ] , x ∈ G , for a random field u : (t , x) ∈ [ 0 , T ] × G ↦ u (t , x) = : u t (x) ∈ R , where Δ α 2 : = − (− Δ) α 2 , α ∈ (0 , 2 ] , is the fractional Laplacian, T ∈ (0 , ∞) is arbitrarily fixed, G ⊂ R d is a bounded domain, g : [ 0 , T ] × G × Ω → R is a joint measurable coefficient, and η t , t ∈ [ 0 , ∞) , is either a Brownian motion or a Lévy process on a given filtered probability space (Ω , F , P ; { F t } t ∈ [ 0 , T ]). To this end, we derive the BMO estimates and Morrey–Campanato estimates, respectively, for stochastic singular integral operators arising from the equations concerned. Then, by utilizing the embedding theory between the Campanato space and the Hölder space, we establish the controllability of the norm of the space C θ , θ / 2 (D ¯) , where θ ≥ 0 , D ¯ = [ 0 , T ] × G ¯. With all these in hand, we are able to show that the q -th order BMO quasi-norm of the α q 0 -order derivative of the solution u is controlled by the norm of g under the condition that η t is a Lévy process. Finally, we derive the Schauder estimate for the p -moments of the solution of the above stochastic fractional heat equations driven by Lévy noise. [ABSTRACT FROM AUTHOR]

We find a pathwise decomposition of a certain class of Brownian semistationary processes (BSS) in terms of fractional Brownian motions. To do this, we specialize in the case when the kernel of the BSS is given by ϕα(x) = L(x)xα with α ∈ (-1/2, 0) ∪ (0, 1/2) and L a continuous function slowly varying at zero. We use this decomposition to study some path properties and derive Itˆo's formula for this subclass of BSS processes. [ABSTRACT FROM AUTHOR]

The aim of this article is to consider the existence of solutions, finiteness of moments, and exponential p-stability of stochastic $\nabla$-dynamic equations on an arbitrary closed subset of $\mathbb{R}$, via Lyapunov functions. This work can be considered as a unification and generalization of works dealing with random difference and stochastic differential equations.

In this paper we consider a mean-field backward stochastic differential equation (BSDE) driven by a Brownian motion and an independent Poisson random measure. Translating the splitting method introduced by Buckdahn et al. (2014) to BSDEs, the existence and the uniqueness of the solution ( Y t , ξ , Z t , ξ , H t , ξ ) , ( Y t , x , P ξ , Z t , x , P ξ , H t , x , P ξ ) of the split equations are proved. The first and the second order derivatives of the process ( Y t , x , P ξ , Z t , x , P ξ , H t , x , P ξ ) with respect to x , the derivative of the process ( Y t , x , P ξ , Z t , x , P ξ , H t , x , P ξ ) with respect to the measure P ξ , and the derivative of the process ( ∂ μ Y t , x , P ξ ( y ) , ∂ μ Z t , x , P ξ ( y ) , ∂ μ H t , x , P ξ ( y ) ) with respect to y are studied under appropriate regularity assumptions on the coefficients, respectively. These derivatives turn out to be bounded and continuous in L 2 . The proof of the continuity of the second order derivatives is particularly involved and requires subtle estimates. This regularity ensures that the value function V ( t , x , P ξ ) ≔ Y t t , x , P ξ is regular and allows to show with the help of a new Itô formula that it is the unique classical solution of the related nonlocal quasi-linear integral-partial differential equation (PDE) of mean-field type. [ABSTRACT FROM AUTHOR]

The numerical solutions of stochastic differential delay equations (SDDEs) under the generalized Khasminskii-type condition were discussed by Mao (Appl. Math. Comput. 217, 5512-5524 2011), and the theory there showed that the Euler-Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in Lp) of the numerical solutions for the SDDEs under this generalized condition. In this paper, we will use the truncated EM method developed by Mao (J. Comput. Appl. Math. 290, 370-384 2015) to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition. [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]

In this paper, we first study the martingale problem in a sublinear expectation space. The critical tool is the Evans–Krylov theorem on regularity properties for solutions of fully nonlinear PDEs. Based on the analysis for the martingale problem and inspired by the rough path theory, we then develop stochastic calculus with respect to a general stochastic process, and derive an Itô type formula and the integration-by-parts formula. Our framework is analytic in that it does not rely on the probabilistic concept of “independence” as in the G -expectation theory. [ABSTRACT FROM AUTHOR]

This paper is concerned with the exponential input-to-state stability for a class of stochastic neural networks with mixed delays. Based on a new Lyapunov-Krasovskii functional, Itô’s formula, Dynkin’s formula and some inequality techniques, some novel sufficient conditions ensuring the exponential input-to-state stability in the mean square for the given stochastic neural networks are derived. Some existing results are extended. Two numerical examples are provided to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a rumor propagation model with L évy noise. We first prove that there exists a positive global solution. Then, the asymptotic behaviors around the rumor‐free equilibrium and rumor‐epidemic equilibrium are obtained. Lastly, simulations verify our results. [ABSTRACT FROM AUTHOR]

The authors consider the problem of the existence, with probability 1, of Green’s function of a stochastic m-point Cauchy problem for a higher-order parabolic equation with white noise perturbations taken with negative values only. Estimates are obtained in spaces of functions whose norm contains the mathematical expectation. [ABSTRACT FROM AUTHOR]

With the methods of H2/H∞-control theory, in the presence of noise we solve the optimization problem for a multiplicative stochastic system with several external disturbances (the multiperturbation problem) and vector Wiener processes with arbitrary intensity matrices. We obtain matrix differential equations of Riccati type, reducing the original optimization problem to solving these equations. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating a class of stochastic nonlinear system with periodic coefficients. Some criteria on existence and uniqueness of periodic solution are established for the stochastic nonlinear system. Finally, a numerical example is given to show the effectiveness and merits of the present results. [ABSTRACT FROM AUTHOR]

The rapid development of the Internet, especially the emergence of the social networks, leads rumor propagation into a new media era. In this paper, we are concerned with a stochastic rumor propagation model. Sufficient conditions for extinction and persistence in the mean of the rumor are established. The threshold between persistence in the mean and extinction of the rumor is obtained. Compared with the corresponding deterministic model, the threshold affected by the white noise is smaller than the basic reproduction number R 0 of the deterministic system. [ABSTRACT FROM AUTHOR]

In this paper, we solve the mean-square exponential input-to-state stability problem for a class of stochastic delayed recurrent neural networks with time-varying coefficients. With the aid of stochastic analysis theory and a Lyapunov-Krasovskii functional, we derive a novel criterion that ensures the given system is mean-square exponentially input-to-state stable. Furthermore, the new criterion generalizes and improves some known results. Finally, two examples and their numerical simulations are provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we first prove that the local time associated with symmetric α -stable processes is of bounded p -variation for any p > 2 α − 1 partly based on Barlow’s estimation of the modulus of the local time of such processes. The fact that the local time is of bounded p -variation for any p > 2 α − 1 enables us to define the integral of the local time ∫ − ∞ ∞ ▿ − α − 1 f ( x ) d x L t x as a Young integral for less smooth functions being of bounded q -variation with 1 ≤ q < 2 3 − α . When q ≥ 2 3 − α , Young’s integration theory is no longer applicable. However, rough path theory is useful in this case. The main purpose of this paper is to establish a rough path theory for the integration with respect to the local times of symmetric α -stable processes for 2 3 − α ≤ q < 4 . [ABSTRACT FROM AUTHOR]

The study of dynamics of epidemics, especially, double epidemics, has got paramount importance in the present days, since various health risks, an individual is exposed to due to the increasing global warming are unavoidable and/or due to limited medical resources available to handle suddenly developed disease. In view of this, we consider a study of an epidemic model with a double hypothesis, namely, SIR and SIRS mechanisms. First, we prove the local asymptotic stability of equilibrium points followed by the global stability of the disease for equilibrium established through the Lyapunov function. Next, we show the existence, extinction and persistence results of the stochastic system. Finally, some numerical examples are presented to support the theoretical results established. • Epidemic model with double hypothesis combined with SIR and SIRS transmission mechanisms. • Global existence of a positive solution. • Global stability of the disease-free equilibrium. • Extinction of the diseases. [ABSTRACT FROM AUTHOR]

We propose a new stochastic competition chemostat system with saturated growth rate and impulsive toxicant input. The main purpose of this paper is to study the stochastic dynamics of a high-dimensional impulsive stochastic chemostat model and find the threshold between persistence and extinction for the impulsive stochastic chemostat system. First, we investigate the stability of the periodic solution of a deterministic impulsive chemostat model and obtain the threshold between persistence and extinction for the system. Second, by using qualitative analysis method of impulsive stochastic differential equations, we obtain conditions for the extinction and persistence in mean of two microorganisms in the stochastic chemostat model. The results show that a stochastic disturbance or the impulsive effect can cause the microorganisms to go to extinction. Finally, we provide some examples together with numerical simulations to illustrate the analytical results and explain the biological implications. [ABSTRACT FROM AUTHOR]

In this paper, we present the threshold of a stochastic SIQS epidemic model which determines the extinction and persistence of the disease. Furthermore, we find that noise can suppress the disease outbreak. Numerical simulations are also carried out to confirm the analytical results. [ABSTRACT FROM AUTHOR]