On the Stability of Stochastic Dynamic Equations on Time Scales
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On the Stability of Stochastic Dynamic Equations on Time Scales Le Anh Tuan1 · Nguyen Thanh Dieu2 · Nguyen Huu Du3
Received: 7 February 2017 / Revised: 11 May 2017 / Accepted: 2 June 2017 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2017
Abstract This paper is concerned with some sufficient conditions ensuring the stochastic stability and the almost sure exponential stability of stochastic differential equations on time scales via Lyapunov functional methods. This work can be considered as a unification and generalization of works dealing with these areas of stochastic difference and differential equations. Keywords Dynamic equations on time scale · Quadratic co-variation · Martingales · Itˆo’s formula · Stochastic exponential function · Lyapunov stability Mathematics Subject Classification (2010) 60H10 · 60J60 · 34A40 · 34D20 · 39A13
1 Introduction The direct method, named also Lyapunov functional method, has become the most widely used tool for studying the exponential stability of stochastic equations. For differential equations, we mention the interesting book of Khas’minskii [11] dealing with necessary
Nguyen Huu Du
[email protected] Le Anh Tuan [email protected] Nguyen Thanh Dieu [email protected] 1
Faculty of Fundamental Science, Hanoi University of Industry, Tu Liem district, Ha Noi, Vietnam
2
Department of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam
3
Faculty of Mathematics, Mechanics, and Informatics, University of Science-VNU, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
L. A. Tuan et al.
and sufficient criterion for almost sure exponential stability of linear Itˆo equation, which opened a new chapter in stochastic stability theory. Since then, many mathematicians have devoted their interests in the theory of stochastic stability. We here mention Arnold [1], Baxendale [2], Kolmanovskii [12], Mohammed [19], Pardoux [20], Pinsky [22], ... Most of these researches were restricted on the study of the stability for the classical Itˆo stochastic differential equations. In 1989, Mao published the papers [15, 16] which can be considered as the first works concerning the stability of stochastic differential equations with respect to semimartingales. For the stability of nonlinear random difference systems, we can refer to [21, 23–25]. On the other hand, 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 stability theory for deterministic dynamic equations on time scales have been investigated for a long history (see [3, 13, 18, 26]), as far as we know, we can only refer to very few papers [4, 8] dealing with the stochastically stability and the almost sure exponential stability of stochastic dynamic equations on time scales. In [8], the authors studied the exponential P -stability of stochastic ∇-dynamic equations on time scales, via Lyapunov function. Continui
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