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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]