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This paper examines the stochastic averaging principle of fast-slow-coupled neutral stochastic differential equations with time-varying delay. Due to the presence of neutral terms, traditional martingale methods and weak convergence techniques are not directly applicable. To overcome these difficulties, this paper gives a more subtle proof for tightness of the slow-varying process. To characterize the limit neutral diffusion system, more complicated computations and more subtle techniques are needed to deal with the neutral term. As a byproduct, this paper also establishes the equivalence between the weak solution of the neutral stochastic differential equation with time-varying delay and the solution of its corresponding the martingale problem. [ABSTRACT FROM AUTHOR]

This work focuses on stochastic functional differential equations (SFDEs) with wide-band noise perturbation, which is a class of actual physical process and has wide application in the modeling of communication and signal system. Using functional derivatives together with martingale methods and weak convergence techniques, this paper examines the asymptotic properties of the underlying systems as the small parameter tends to zero. Based on the techniques, this paper also establishes the average principle for a class of SFDEs with two-time scales. As a special case, this paper also considers a class of integro-differential equations with wide-band noise perturbation and examines its approximate properties. As an example, a scalar stochastic Lotka–Volterra integro-differential system with wide-band noise perturbation is investigated. [ABSTRACT FROM AUTHOR]

This paper develops strong convergence of the Euler–Maruyama (EM) schemes for approximating McKean–Vlasov stochastic differential equations (SDEs). In contrast to the existing work, a novel feature is the use of a much weaker condition—local Lipschitzian in the state variable, but under uniform linear growth assumption. To obtain the desired approximation, the paper first establishes the existence and uniqueness of solutions of the original McKean–Vlasov SDE using a Euler-like sequence of interpolations and partition of the sample space. Then, the paper returns to the analysis of the EM scheme for approximating solutions of McKean–Vlasov SDEs. A strong convergence theorem is established. Moreover, the convergence rates under global conditions are obtained. [ABSTRACT FROM AUTHOR]

Using Malliavin calculus, this paper establishes asymptotic Bismut formulae for stochastic functional differential equations with infinite delay. Both nondegenerate and degenerate diffusion coefficients are treated. In addition, combined with the corresponding exponential ergodicity, stabilization bounds for \nabla P_{t}f as t\rightarrow \infty are derived. [ABSTRACT FROM AUTHOR]

This paper considers stochastic functional differential equations (SFDEs) with infinite delay. The main aim is to establish the LaSalle-type theorems to locate limit sets for this class of SFDEs. In comparison with the existing results, this paper gives more general results under the weaker conditions imposed on the Lyapunov function. These results can be used to discuss the asymptotic stability and asymptotic boundedness for SFDEs with infinite delay. In the end, two examples will be given to illustrate applications of our new results established. [ABSTRACT FROM AUTHOR]

Gene regulatory networks, which are complex high-dimensional stochastic dynamical systems, are often subject to evident intrinsic fluctuations. It is deemed reasonable to model the systems by the chemical Langevin equations. Since the mRNA dynamics are faster than the protein dynamics, we have a two-time scales system. In general, the process of protein degradation involves time delays. In this paper, we take the system memory into consideration in which we consider a model with a complete memory represented by an integral delay from 0 to t. Based on the averaging principle and perturbed test function method, this work examines the weak convergence of the slow-varying process. By treating the fast-varying process as a random noise, under appropriate conditions, it is shown that the slow-varying process converges weakly to the solution of a stochastic differential delay equation whose coefficients are the average of those of the original slow-varying process with respect to the invariant measure of the fast-varying process. [ABSTRACT FROM AUTHOR]

In stochastic financial and biological models, the diffusion coefficients often involve the term , or more general |x|r, r ∈ (0, 1). These coefficients do not satisfy the local Lipschitz condition, which implies that the existence and uniqueness of the solution cannot be obtained by the standard conditions. This article establishes the existence of the weak solution for this class of stochastic differential equations by using the martingale representation and weak convergence methods. [ABSTRACT FROM AUTHOR]

By a semi-discrete tamed Euler approximation as a bridge, this paper examines the tamed Euler scheme for the Mckean-Vlasov stochastic differential equations (SDEs) and obtains the convergence rate under the one-sided Lipschitz condition for the drift coefficient. Under the local Lipschitz condition for the drift coefficient, the strong convergence of the tamed scheme is also examined. When the coefficients satisfy the Lipschitz condition, this paper also obtains the asymptotic error distribution. [ABSTRACT FROM AUTHOR]

Based on the martingale theory and large deviation techniques, we investigate the pth moment exponential stability criterion of the exact and numerical solutions to hybrid stochastic differential equations (SDEs) under the local Lipschitz condition. This new stability criterion shows that Markovian switching can serve as a stochastic stabilizing factor by its logarithmic moment-generating function. We also investigate the pth moment exponential stability of Euler–Maruyama (EM), backward EM (BEM) and split-step backward EM (SSBEM) approximations for hybrid SDEs and show that, under the additional linear growth condition, the EM method can share the mean-square exponential stability of the exact solution for sufficiently small step size. However, the BEM method can work without the linear growth condition. We further investigate the SSBEM method under a coupled condition. [ABSTRACT FROM PUBLISHER]

In this study, the distributed averaging of high‐dimensional first‐order agents is investigated with relative‐state‐dependent measurement noises. Each agent can measure or receive its neighbours' state information with random noises, whose intensity is a non‐linear matrix function of agents' relative states. By the tools of stochastic differential equations and algebraic graph theory, the authors give sufficient conditions to ensure mean square and almost sure average consensus and the convergence rate and the steady‐state error for average consensus are quantified. Especially, if the noise intensity function depends linearly on the relative distance of agents' states, then a sufficient condition is given in terms of the control gain, the noise intensity coefficient constant, the number of agents and the dimension of agents' dynamics. [ABSTRACT FROM AUTHOR]

This paper considers a given neutral functional system under regime switching whose solution grows at most polynomially. Can the system tolerate Brownian noise and colour noise to preserve the polynomial growth if the system is subject to another environment noise? The answer is positive. This paper shows that environmental noise and regime switching work together to make the original neutral differential system whose solution grows at most polynomially become a new neutral functional differential system whose solution will still preserve polynomial growth. This indicates that this neutral functional differential system can tolerate small noise without losing the property of polynomial growth, which implies robustness. This paper also proves that the criterion is suitable to neutral delay system which is as a special case of functional system. Finally, two examples are given to illustrate the main theory. [ABSTRACT FROM AUTHOR]

This work focuses on gene regulatory networks driven by intrinsic noise with two-time scales. It uses a stochastic averaging approach for these systems to reduce complexity. Comparing with the traditional quasi-steady-state hypothesis (QSSH), our approach uses stochastic averaging principle to treat the intrinsic noise coming from both the fast-changing variables and the slow-changing variables, which yields a more precise description of the underlying systems. To provide further insight, this paper also investigates a prototypical two-component activator-repressor genetic circuit model as an example. If all the protein productions were linear, these two methods would yield the same reduction result. However, if one of the protein productions is nonlinear, the stochastic averaging principle leads to a different reduction result from that of the traditional QSSH. [ABSTRACT FROM AUTHOR]

This paper examines exponential mean square stability of the split-step theta approximation and the stochastic theta method for the stochastic differential delay equations and stochastic ordinary differential equations (SODEs) under a coupled monotone condition on drift and diffusion coefficients. It is shown that forthe two classes of the theta approximations can preserve the exponential mean square stability when some conditions on the stepsize and drift coefficient are imposed, but for, without the globally Lipschitz continuity, these two classes of theta methods show exponentially mean square stability unconditionally. Moreover, for sufficiently small stepsize, the decay rate as measured by the bound of the Lyapunov exponent can be reproduced arbitrarily accurately. Some results in this paper extend the existing results for linear SODEs to nonlinear stochastic differential equations (SDEs), and also improve our previous results of numerical stability of nonlinear SDEs. [ABSTRACT FROM AUTHOR]

This work studies stability and stochastic stabilization of numerical solutions of a class of regime-switching jump diffusion systems. These systems have a wide range of applications in communication systems, flexible manufacturing and production planning, financial engineering and economics because they involve three classes of stochastic factors: white noise, Poisson jump and Markovian switching. This paper focuses on the stability of numerical solutions of the switching jump diffusion systems and examines the conditions under which the Euler–Maruyama (EM) and the backward EM may share the stability of the exact solution. These conditions show that all these three classes of stochastic factors may serve as stabilizing factors and play positive roles for the stability property of both exact and numerical solutions. [ABSTRACT FROM PUBLISHER]

The second author's work [F. Wu, X. Mao, and L. Szpruch, Almost sure exponential stability of numerical solutions for stochastic delay differential equations, Numer. Math. 115 (2010), pp. 681–697] and Mao's papers [D.J. Higham, X. Mao, and C. Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal. 45 (2007), pp. 592–607; X. Mao, Y. Shen, and G. Alison, Almost sure exponential stability of backward Euler–Maruyama discretizations for hybrid stochastic differential equations, J. Comput. Appl. Math. 235 (2011), pp. 1213–1226] showed that the backward Euler–Maruyama (BEM) method may reproduce the almost sure stability of stochastic differential equations (SDEs) without the linear growth condition of the drift coefficient and the counterexample shows that the Euler–Maruyama (EM) method cannot. Since the stochastic θ-method is more general than the BEM and EM methods, it is very interesting to examine the interval in which the stochastic θ-method can capture the stability of exact solutions of SDEs. Without the linear growth condition of the drift term, this paper concludes that the stochastic θ-method can capture the stability for θ∈(1/2, 1]. For θ∈[0, 1/2), a counterexample shows that the stochastic θ-method cannot reproduce the stability of the exact solution. Finally, two examples are given to illustrate our conclusions. [ABSTRACT FROM PUBLISHER]

In the previous work by the second author, it was shown that the polynomial Brownian noise may suppress the potential explosion of the nonlinear system without the linear growth condition or the one-sided linear growth condition and the linear Brownian noise may stabilise this suppressed system. This article is a continuation of our previous paper and considers the asymptotic properties of the suppressed system. These asymptotic properties show that the suppressed system by polynomial Brownian noise will grow with at most polynomial speed. [ABSTRACT FROM PUBLISHER]

In this paper, a class of jump systems with the mean-reverting γ-process are considered in finance. The analytical properties including the positivity, boundedness and pathwise estimations of the solution are discussed. Moreover, the authors show that the Euler-Maruyama approximate solutions converge to the true solutions in probability. Finally, the authors apply the convergence to examine a bond and a path-dependent option price in the financial pricing. [ABSTRACT FROM AUTHOR]

Without the linear growth condition on the drift coefficient, this article examines the existence and uniqueness of global solutions of a class of neutral stochastic differential equations with unbounded delay and their asymptotic stabilities with general decay rate. To illustrate the application of our results, this article gives a two-dimensional system as an example. [ABSTRACT FROM AUTHOR]

This article considers a class of nonlocal stochastic functional differential equations with infinite delay whose coefficients are dependent the pth moment and establishes the existence-and-uniqueness theorem under the conditions that are similar to the classical linear growth condition and the Lipschitz condition. Compared with the existing results, the conditions of this article are easier to test. [ABSTRACT FROM AUTHOR]

By the continuous and discrete nonnegative semimartingale convergence theorems, this paper investigates conditions under which the Euler--Maruyama (EM) approximations of stochastic functional differential equations (SFDEs) can share the almost sure exponential stability of the exact solution. Moreover, for sufficiently small stepsize, the decay rate as measured by the Lyapunov exponent can be reproduced arbitrarily accurately. [ABSTRACT FROM AUTHOR]

The mean-reverting square root process with jump has been widely used as a model on the financial market. Since the diffusion coefficient in the model does not satisfy the linear growth condition and local Lipschitz condition, we can not examine its properties by traditional techniques. To overcome the difficulties, we develop several new techniques to examine the numerical method of jump models involving delay and mean-reverting square root. We show that the numerical approximate solutions converge to the true solutions. Finally, we apply the convergence to examine a path-dependent option price and a bond in the financial pricing. [ABSTRACT FROM AUTHOR]

In general, time delay and system uncertainty are commonly encountered for all population systems. To examine whether the presence of environmental noise affects infinite delay population systems significantly, this paper perturbs the functional Kolmogorov-type system with infinite delay [image omitted] into the infinite delay stochastic functional differential system [image omitted] By the M-matrix technique, this paper examines the global positive solution and its pathwise estimation for this stochastic functional Kolmogorov-type population system. To illustrate the applications of our theory more clearly, this paper also discusses a Lotka-Volterra system with mixed delays as a special case. [ABSTRACT FROM AUTHOR]

In general, population systems are often subject to environmental noise. To examine whether the presence of such noise affects these systems significantly, we perturb the functional Kolmogorov-type system$$\dot{x}(t)=\mathrm{diag}(x_1(t),\dots,x_n(t))f(x_t)$$into the stochastic functional differential equation$$\mathrm{d} x(t)=\mathrm{diag}(x_1(t),\dots,x_n(t))[f(x_t)\,\mathrm{d} t+g(x_t)\,\mathrm{d} w(t)].$$We show that different environmental noise structures have different effects on the population system with unbounded delay. Under two classes of different environmental noise perturbations, we establish existence theorems of the global positive solution to the unbounded delay stochastic functional Kolmogorov-type system. As the desired results for population dynamics, we also examine asymptotic boundedness, including the moment boundedness, stochastically ultimate boundedness and the moment average boundedness in time. To illustrate our idea more clearly, as a special case we also discuss a Lotka–Volterra system with unbounded delay. [ABSTRACT FROM PUBLISHER]

This paper establishes the existence-and-uniqueness theorems of the global solution to a class of stochastic delay differential equations only with the local Lipschitz condition but does not consider the linear growth condition and studies stochastic ultimate boundedness and the moment average in time. Here we allow the coefficients of these equations to be polynomial or their growths are controlled by polynomial speed. To illustrate our idea clearly, we consider a scalar equation and a special n-dimensional equation carefully. [ABSTRACT FROM AUTHOR]

This article deals with a class of generalized stochastic delay Lotka-Volterra systems of the form dX(t) = diag(X1(t), X2(t),..., Xn(t))[(f(X(t)) + g(X(t - τ)))dt + h(X(t))dB(t)]. Under some unrestrictive conditions on f, g, and h, we show that the unique solution of such a stochastic system is positive and does not explode in a finite time with probability one. We also establish some asymptotic boundedness results of the solution including the time average of its (β + α)-order moment, as well as its asymptotic pathwise estimation. As a by-product, a stochastic ultimate boundedness of the solution for this stochastic system is directly derived. Three examples are given to illustrate our conclusions. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to give some new conditions under which a class of nonlinear stochastic differential delay equations with Markovian switching admit a unique solution and this solution has nice asymptotic properties, including the moment boundedness and the moment boundedness average in time of this solution. These new conditions show that the coefficients of the nonlinear stochastic differential delay equation with Markovian switching are polynomial or controlled by the polynomial functions or functionals. Two examples are also given for illustration. [ABSTRACT FROM AUTHOR]

This paper investigates the existence and boundedness of the global positive solutions for stochastic Kolmogorov system with infinite delay. We obtain two classes of sufficient conditions to guarantee these properties. These two classes of the conditions show that the system is dominated by deterministic part or stochastic part respectively. By the M-matrix technique, we conveniently compare our results. Finally, a Lotka–Volterra model as a special case is given to illustrate our idea. [ABSTRACT FROM AUTHOR]

In general, population systems are often subject to environmental noise. To examine whether the presence of such noise affects population systems significantly, we stochastically perturb the delay Kolmogorov-type system [image omitted] = diag(x1(t),..., xn(t))f(x(t), x(t - δ(t))) into the stochastic delay differential equation dx(t) = diag(x1(t),..., xn(t))[f(x(t), x(t - δ(t)))dt + g(x(t), x(t - δ(t)))dw(t)]. Under the traditional diagonal dominance condition, we study the existence and uniqueness of the global positive solution of this stochastic system, and its asymptotic-bound properties. These properties are natural requirements from the biological point of view. As the special cases, we also discuss some stochastic Lotka-Volterra systems. [ABSTRACT FROM AUTHOR]