On Halanay-type analysis of exponential stability for the θ-Maruyama method for stochastic delay differential equations

C. T H Baker, Evelyn Buckwar

    Research output: Contribution to journalArticlepeer-review

    Abstract

    Using an approach that has its origins in work of Halanay, we consider stability in mean square of numerical solutions obtained from the θ-Maruyama discretization of a test stochastic delay differential equation dX(t) = {f(t) - αX(t) + βX(t - τ)}dt + {g(t) + η X(t) + μX(t - τ)} dW(t), interpreted in the Itô sense, where W(t) denotes a Wiener process. We focus on demonstrating that we may use techniques advanced in a recent report by Baker and Buckwar to obtain criteria for asymptotic and exponential stability, in mean square, for the solutions of the recurrence X̃n+1 - X̃n = θh{fn+1 - αX̃n+1 + βX̃n+1-N} + (1 - θ)h{fn - αX̃n + βX̃n-N} + √h(gn + ηX̃n + μX̃n-N}ξn, ξn ε N (0,1). © World Scientific Publishing Company.
    Original languageEnglish
    Pages (from-to)201-209
    Number of pages8
    JournalStochastics and Dynamics
    Volume5
    Issue number2
    DOIs
    Publication statusPublished - Jun 2005

    Keywords

    • θ-Maruyama scheme
    • Asymptotic and exponential stability
    • Halanay-type inequalities
    • Stochastic delay differential & difference equations

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