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Xuerong Mao - One of the best experts on this subject based on the ideXlab platform.

  • stability equivalence between the stochastic Differential Delay equations driven by g brownian motion and the euler maruyama method
    Applied Mathematics Letters, 2019
    Co-Authors: Chen Fei, Shounian Deng, Weiyin Fei, Xuerong Mao
    Abstract:

    Abstract Consider a stochastic Differential Delay equation driven by G -Brownian motion ( G -SDDE) d x ( t ) = f ( x ( t ) , x ( t − τ ) ) d t + g ( x ( t ) , x ( t − τ ) ) d B ( t ) + h ( x ( t ) , x ( t − τ ) ) d 〈 B 〉 ( t ) . Under the global Lipschitz condition for the G -SDDE, we show that the G -SDDE is exponentially stable in mean square if and only if for sufficiently small step size, the Euler–Maruyama (EM) method is exponentially stable in mean square. Thus, we can carry out careful numerical simulations to investigate the exponential stability of the underlying G -SDDE in practice, in the absence of an appropriate Lyapunov function. A numerical example is provided to illustrate our results.

  • the truncated euler maruyama method for stochastic Differential Delay equations
    arXiv: Numerical Analysis, 2017
    Co-Authors: Qian Guo, Xuerong Mao, Rongxian Yue
    Abstract:

    The numerical solutions of stochastic Differential Delay equations (SDDEs) under the generalized Khasminskii-type condition were discussed by Mao [15], and the theory there showed that the Euler-Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in L^p) of the numerical solutions for the SDDEs under this generalized condition. In this paper, we will use the truncated EM method developed by Mao [16] to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition.

  • robust stability and boundedness of nonlinear hybrid stochastic Differential Delay equations
    IEEE Transactions on Automatic Control, 2013
    Co-Authors: Xuerong Mao, Liguo Zhang
    Abstract:

    One of the important issues in the study of hybrid stochastic Differential Delay equations (SDDEs) is the automatic control, with consequent emphasis being placed on the asymptotic analysis of stability and boundedness. In the study of asymptotic properties, the robust stability has received a great deal of attention. The theory of robust stability shows how much perturbation a given stable hybrid SDDE can tolerate so that its perturbed system remains stable. Almost all results so far on the robust stability require that the underlying SDDEs be either linear or nonlinear with linear growth condition. However, little is known on the robust stability of nonlinear hybrid SDDEs without the linear growth condition, which is one of the key topics in this paper. The other key topic is the robust boundedness. The aim here is to answer the question: how much perturbation can a given asymptotically bounded hybrid SDDE tolerate so that its perturbed system remains asymptotically bounded?

  • stability and boundedness of nonlinear hybrid stochastic Differential Delay equations
    Systems & Control Letters, 2013
    Co-Authors: Xuerong Mao, Yi Shen
    Abstract:

    Abstract One of the important issues in the study of hybrid SDDEs is the automatic control, with consequent emphasis being placed on the asymptotic analysis of stability and boundedness (see e.g.  [5] , [10] , [11] , [13] , [14] , [15] , [17] , [19] , [21] ). The method of Lyapunov functions is one of the most powerful techniques in the study of stability and boundedness. So far, most of the results in this area do not only require the Lyapunov functions in different modes have the same feature (e.g. polynomials with the same degree) but also that the diffusion operator in different modes be bounded by the same type of functions. These requirements are restrictive and often cannot be met by those hybrid SDDEs that have different nonlinear structures in different modes. To study the stability and boundedness of such hybrid SDDEs, we will in this paper use different types of Lyapunov functions (e.g. polynomials with different degrees) for different modes. Moreover, the condition on the diffusion operator is relaxed significantly.

  • stochastic Differential Delay equations of population dynamics
    Journal of Mathematical Analysis and Applications, 2005
    Co-Authors: Xuerong Mao, Chenggui Yuan, Jiezhong Zou
    Abstract:

    In this paper, we investigate the almost surely asymptotic stability for the nonlinear stochastic Differential Delay equations with Markovian switching. Some sufficient criteria on the controllability and robust stability are also established for linear stochastic Differential Delay equations with Markovian switching.

Yi Shen - One of the best experts on this subject based on the ideXlab platform.

  • stability and boundedness of nonlinear hybrid stochastic Differential Delay equations
    Systems & Control Letters, 2013
    Co-Authors: Xuerong Mao, Yi Shen
    Abstract:

    Abstract One of the important issues in the study of hybrid SDDEs is the automatic control, with consequent emphasis being placed on the asymptotic analysis of stability and boundedness (see e.g.  [5] , [10] , [11] , [13] , [14] , [15] , [17] , [19] , [21] ). The method of Lyapunov functions is one of the most powerful techniques in the study of stability and boundedness. So far, most of the results in this area do not only require the Lyapunov functions in different modes have the same feature (e.g. polynomials with the same degree) but also that the diffusion operator in different modes be bounded by the same type of functions. These requirements are restrictive and often cannot be met by those hybrid SDDEs that have different nonlinear structures in different modes. To study the stability and boundedness of such hybrid SDDEs, we will in this paper use different types of Lyapunov functions (e.g. polynomials with different degrees) for different modes. Moreover, the condition on the diffusion operator is relaxed significantly.

  • taylor approximation of the solutions of stochastic Differential Delay equations with poisson jump
    Communications in Nonlinear Science and Numerical Simulation, 2011
    Co-Authors: Feng Jiang, Yi Shen, Lei Liu
    Abstract:

    Abstract In this paper, we are concerned with the stochastic Differential Delay equations with Poisson jump (SDDEsPJ). As stochastic Differential equations, most SDDEsPJ cannot be solved explicitly. Therefore, numerical solutions have become an important issue in the study of SDDEsPJ. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEsPJ when the drift and diffusion coefficients are Taylor approximations.

  • almost surely asymptotic stability of neutral stochastic Differential Delay equations with markovian switching
    Stochastic Processes and their Applications, 2008
    Co-Authors: Yi Shen, Chenggui Yuan
    Abstract:

    The main aim of this paper is to discuss the almost surely asymptotic stability of the neutral stochastic Differential Delay equations (NSDDEs) with Markovian switching. Linear NSDDEs with Markovian switching and nonlinear examples will be discussed to illustrate the theory.

Wu Hao - One of the best experts on this subject based on the ideXlab platform.

  • anticipated backward stochastic Differential equations with jumps under the non lipschitz condition
    Statistics & Probability Letters, 2014
    Co-Authors: Wu Hao
    Abstract:

    Abstract This paper deals with a class of anticipated backward stochastic Differential equations with Poisson jumps (ABSDEJs). We first show that there is a duality between anticipated backward stochastic Differential equations with jumps and stochastic Differential Delay equations with jumps (SDDEJs). Then, we prove the existence and uniqueness of adapted solutions and L p solutions for such ABSDEJs under the non-Lipschitz conditions as well as a comparison theorem is obtained through constructing some iterative equations which are different from iterative equations in Peng and Yang (2009).

Chenggui Yuan - One of the best experts on this subject based on the ideXlab platform.

Liguo Zhang - One of the best experts on this subject based on the ideXlab platform.

  • robust stability and boundedness of nonlinear hybrid stochastic Differential Delay equations
    IEEE Transactions on Automatic Control, 2013
    Co-Authors: Xuerong Mao, Liguo Zhang
    Abstract:

    One of the important issues in the study of hybrid stochastic Differential Delay equations (SDDEs) is the automatic control, with consequent emphasis being placed on the asymptotic analysis of stability and boundedness. In the study of asymptotic properties, the robust stability has received a great deal of attention. The theory of robust stability shows how much perturbation a given stable hybrid SDDE can tolerate so that its perturbed system remains stable. Almost all results so far on the robust stability require that the underlying SDDEs be either linear or nonlinear with linear growth condition. However, little is known on the robust stability of nonlinear hybrid SDDEs without the linear growth condition, which is one of the key topics in this paper. The other key topic is the robust boundedness. The aim here is to answer the question: how much perturbation can a given asymptotically bounded hybrid SDDE tolerate so that its perturbed system remains asymptotically bounded?