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This paper studies a class of McKean-Vlasov stochastic functional differential equations in which the dynamics of the underlying process may depend on its history as well as the distribution of the history. These stochastic processes provide a general and versatile framework for modeling and analyzing complex systems with inherent distribution- and memory-dependent dynamical behaviors. Under mild conditions, this paper establishes the well-posedness for the associated McKean-Vlasov stochastic functional differential equations and derives a propagation of chaos as well as a continuity with respect to the initial distribution result. Several examples are studied as an illustration. [ABSTRACT FROM AUTHOR]

This paper examines the stationary distribution of the stochastic Lotka-Volterra model with infinite delay. Since the solutions of stochastic functional or delay differential equations depend on their history, they are non-Markov, which implies that traditional techniques based on the Markov property, cannot be applicable. This paper uses the variable substituting technique to obtain a higher-dimensional stochastic differential equation without delay. In this paper it is shown that the new equation satisfies the strong Feller property and strong Markov property, by which the invariant measure follows from the Krylov-Bogoliubov Theorem. Since distribution of the original model is actually the marginal distribution of the new equation, these results also show that there exists the stationary distribution for the original equation. Moreover, when the noise intensity is strongly dependent on the population size, by Hörmander's theorem, this paper also shows that the stationary distribution of this stochastic Lotka-Volterra system with infinite delay holds a C ∞ -smooth density. [ABSTRACT FROM AUTHOR]

This paper focuses on two-time-scale coupled stochastic functional differential equations (SFDEs). The system under consideration has a slow component and a fast component. Both components depend on the segment process (an infinite dimension process) of the slow component. To overcome the difficulty due to the past dependence and the coupling of the segment process, such properties as the Hölder continuity and tightness on a space of continuous functions are investigated first for the segment process. In addition, it is also shown that the solution of a fixed- x equation depends continuously on the parameters. Then using the martingale problem formulation, an average principle is established by a direct averaging. [ABSTRACT FROM AUTHOR]

This paper examines limit theorems for a class of stochastic functional differential equations with infinite delay. Under non-Lipschitz conditions, the strong law of large numbers and the central limit theorem for additive functionals of the segment processes are established by using limit theorems of uniform mixing Markovian processes. Under some dissipative assumptions, the law of iterated logarithm is also derived for additive functionals of such equations via a martingale difference sequence of square integrable martingales. [ABSTRACT FROM AUTHOR]

Delays are ubiquitous, pervasive, and entrenched in everyday life, thus taking it into consideration is necessary. Dupire recently developed a functional Itô formula, which has changed the landscape of the study of stochastic functional differential equations and encouraged a reconsideration of many problems and applications. Based on the new development, this work examines functional diffusions with two-time scales in which the slow-varying process includes path-dependent functionals and the fast-varying process is a rapidly-changing diffusion. The gene expression of biochemical reactions occurring in living cells in the introduction of this paper is such a motivating example. This paper establishes mixed functional Itô formulas and the corresponding martingale representation. Then it develops an averaging principle using weak convergence methods. By treating the fast-varying process as a random "noise", under appropriate conditions, it is shown that the slow-varying process converges weakly to a stochastic functional differential equation whose coefficients are averages of that of the original slow-varying process with respect to the invariant measure of the fast-varying process. [ABSTRACT FROM AUTHOR]

This work is devoted to stochastic functional differential equations (SFDEs) with infinite delay. First, existence and uniqueness of the solutions of such equations are examined. Because the solutions of the delay equations are not Markov, a viable alternative for studying further asymptotic properties is to use solution maps or segment processes. By examining solution maps, this work investigates the Markov properties as well as the strong Markov properties. Also obtained are adaptivity and continuity, mean-square boundedness, and convergence of solution maps from different initial data. This paper then examines the ergodicity of underlying processes and establishes existence of the invariant measure for SFDEs with infinite delay under suitable conditions. [ABSTRACT FROM AUTHOR]

Abstract: Populations of biological species are often subject to different types of environmental noises. These noises play crucial roles and have significant impact on the evolution and biodiversity. To understand the effects of different types of noises on the asymptotic properties of the populations, this paper reveals the influence of inherent net birth noise and the interaction noise among stochastically perturbed population systems. By considering systems with infinite delays, (1) this paper discovers that (a) the interaction noise may suppress the potential explosion and guarantee the existence of the global solution, which yields regularity of the stochastic system, and (b) the inherent net birth noise may lead to extinction of the populations; (2) this paper provides sufficient conditions that ensure the stochastic persistence, stochastic boundedness, and asymptotic polynomial growth of the population system; (3) this paper demonstrates that these asymptotic properties are completely determined by stochastic noises and are independent of other parameters. [Copyright &y& Elsevier]

Abstract: This work examines almost sure stability of a pure random delay system whose delay time is modeled by a finite state continuous-time Markov chain with two-time scales. The Markov chain contains a fast-varying part and a slowly-changing part. Using the properties of the weighted occupation measure of the Markov chain, it is shown that the overall systemʼs almost-sure-asymptotic stability can be obtained by using the “averaged” delay. This feature implies that even if some longer delay times may destabilize the system individually, the system may still be stable if their impact is balanced. In other words, the Markov chain becomes a stabilizing factor. Numerical results are provided to demonstrate our results. [Copyright &y& Elsevier]

Abstract: This paper investigates the asymptotic behavior of the stochastic Lotka–Volterra model with multiple delays. By using certain matrix techniques, some results on nonexplosion, boundedness and pathwise estimation of the model are investigated. [Copyright &y& Elsevier]

Abstract: In general, population systems are often subject to environmental noise. This paper considers the stochastic functional Kolmogorov-type system Under the traditionally diagonally dominant condition, we study existence and uniqueness of the global positive solution of this stochastic system, and its asymptotic bound properties and moment average boundedness in time. These properties are natural requirements from the biological point of view. As the special cases, we also discuss some stochastic Lotka–Volterra systems. [Copyright &y& Elsevier]

Abstract: Empirical studies show that the most successful continuous-time models of the short-term rate in capturing the dynamics are those that allow the volatility of interest changes to be highly sensitive to the level of the rate. However, from the mathematics, the high sensitivity to the level implies that the coefficients do not satisfy the linear growth condition, so we can not examine its properties by traditional techniques. This paper overcomes the mathematical difficulties due to the nonlinear growth and examines its analytical properties and the convergence of numerical solutions in probability. The convergence result can be used to justify the method within Monte Carlo simulations that compute the expected payoff of financial products. For illustration, we apply our results compute the value of a bond with interest rate given by the highly sensitive mean-reverting process as well as the value of a single barrier call option with the asset price governed by this process. [Copyright &y& Elsevier]

Abstract: In this paper we stochastically perturb the functional Kolmogorov-type system into the stochastic functional differential equation This paper studies existence and uniqueness of the global positive solution, and its asymptotic bound properties and moment average in time. These properties are natural requirements from the biological point of view. As the special cases, we discuss the various stochastic Lotka–Volterra systems. [Copyright &y& Elsevier]

This work focuses on functional differential equations subject to wideband noise perturbations. Modeling using a white noise is often an idealization of the actual physical process, whereas a wideband noise can be easily realized in applications and well approximates a white noise. Using functional derivatives together with the combined perturbed test function methods and martingale techniques, this paper demonstrates that when a small parameter tends to zero, the underlying process converges to a limit that is the solution of a stochastic functional differential equation. To illustrate, an integro-differential system with wideband noise perturbation is examined as an example. Not only are the results interesting from a mathematical point of view, but also they are of utility to a wide range of applications. [ABSTRACT FROM AUTHOR]

This work focuses on a class of stochastic damping Hamiltonian systems with state-dependent switching, where the switching process has a countably infinite state space. After establishing the existence and uniqueness of a global weak solution via the martingale approach under very mild conditions, the paper next proves the strong Feller property for regime-switching stochastic damping Hamiltonian systems by the killing technique together with some resolvent and transition probability identities. The commonly used continuity assumption for the switching rates q k l (⋅) in the literature is relaxed to measurability in this paper. Finally the paper provides sufficient conditions for exponential ergodicity and large deviations principle for regime-switching stochastic damping Hamiltonian systems. Several examples on regime-switching van der Pol and (overdamped) Langevin systems are studied in detail for illustration. [ABSTRACT FROM AUTHOR]

This paper is concerned with approximations of invariant probability measures for stochastic functional differential equations (SFDEs) using a backward Euler-Maruyama (BEM) scheme under one-sided Lipschitz condition on the drift coefficient. Firstly, the strong convergence of the numerical "segment sequence" from the BEM scheme on finite time interval [ 0 , T ] is established. In addition, it is also demonstrated that the numerical segment sequence from the BEM scheme is a Markov process, and the corresponding discrete-time semigroup generated by this BEM scheme admits a unique numerical invariant probability measure. Finally, it is revealed that the numerical invariant probability measure converges to the underlying one in a Wasserstein distance. [ABSTRACT FROM AUTHOR]