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Almost Sure Exponential Stability of Stochastic Differential Delay Equations on Time Scales

The aim of this paper is to study the almost sure exponential stability of stochastic differential delay equations on time scales. This work can be considered as a unification and generalization of stochastic difference and stochastic differential delay equations. | VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 3 (2016) 64-75 Almost Sure Exponential Stability of Stochastic Differential Delay Equations on Time Scales Le Anh Tuan* Faculty of Fundamental Science, Hanoi University of Industry, Tu Liem, Hanoi, Vietnam Received 16 August 2016 Revised 15 September 2016; Accepted 09 September 2016 Abstract: The aim of this paper is to study the almost sure exponential stability of stochastic differential delay equations on time scales. This work can be considered as a unification and generalization of stochastic difference and stochastic differential delay equations. Keywords: Delay equation, almost sure exponential stability, Ito formula, Lyapunov function. 1. Introduction The stochastic differential/difference delay equations have come to play an important role in describing the evolution of eco-systems in random environment, in which the future state depends not only on the present state but also on its history. Therefore, their qualitative and quantitative properties have received much attention from many research groups (see [1, 2] for the stochastic differential delay equations and [3-6] for the stochastic difference one). In order to unify the theory of differential and difference equations into a single set-up, the theory of analysis on time scales has received much attention from many research groups. While the deterministic dynamic equations on time scales have been investigated for a long history (see [7-11]), as far as we know, we can only refer to very few papers [12-15] which contributed to the stochastic dynamics on time scales. Moreover, there is no work dealing with the stochastic dynamic delay equations. Recently, in [14], we have studied the exponential p -stability of stochastic -dynamic equations on time scale, via Lyapunov function. Continuing the idea of this article [14], we study the almost sure exponential stability of stochastic dynamic delay equations on time scales. Motivated by the .