The moment exponential stability criterion of nonlinear hybrid stochastic differential equations and its discrete approximations

2016 ◽  
Vol 146 (6) ◽  
pp. 1303-1328 ◽  
Author(s):  
Xiaofeng Zong ◽  
Fuke Wu ◽  
Chengming Huang
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Growth Condition ◽  
Stability Criterion ◽  
Linear Growth ◽  
Numerical Solutions ◽  
Large Deviation ◽  
Linear Growth Condition ◽  
Moment Exponential Stability

Based on the martingale theory and large deviation techniques, we investigate the pth moment exponential stability criterion of the exact and numerical solutions to hybrid stochastic differential equations (SDEs) under the local Lipschitz condition. This new stability criterion shows that Markovian switching can serve as a stochastic stabilizing factor by its logarithmic moment-generating function. We also investigate the pth moment exponential stability of Euler–Maruyama (EM), backward EM (BEM) and split-step backward EM (SSBEM) approximations for hybrid SDEs and show that, under the additional linear growth condition, the EM method can share the mean-square exponential stability of the exact solution for sufficiently small step size. However, the BEM method can work without the linear growth condition. We further investigate the SSBEM method under a coupled condition.

scholarly journals Numerical Solutions of Stochastic Differential Equations with Piecewise Continuous Arguments under Khasminskii-Type Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Minghui Song ◽  
Ling Zhang
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Growth Condition ◽  
Linear Growth ◽  
Numerical Solutions ◽  
Type Theorem ◽  
Global Existence Of Solutions ◽  
Linear Growth Condition ◽  
Piecewise Continuous ◽  
Piecewise Continuous Arguments

The main purpose of this paper is to investigate the convergence of the Euler method to stochastic differential equations with piecewise continuous arguments (SEPCAs). The classical Khasminskii-type theorem gives a powerful tool to examine the global existence of solutions for stochastic differential equations (SDEs) without the linear growth condition by the use of the Lyapunov functions. However, there is no such result for SEPCAs. Firstly, this paper shows SEPCAs which have nonexplosion global solutions under local Lipschitz condition without the linear growth condition. Then the convergence in probability of numerical solutions to SEPCAs under the same conditions is established. Finally, an example is provided to illustrate our theory.

scholarly journals On the existence of solutions for stochastic differential equations driven by fractional Brownian motion

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1695-1700
Author(s):  
Zhi Li
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Growth Condition ◽  
Linear Growth ◽  
Existence Of Solutions ◽  
Hurst Parameter ◽  
Linear Growth Condition ◽  
Discontinuous Drift

In this paper, we are concerned with a class of stochastic differential equations driven by fractional Brownian motion with Hurst parameter 1/2 < H < 1, and a discontinuous drift. By approximation arguments and a comparison theorem, we prove the existence of solutions to this kind of equations under the linear growth condition.

scholarly journals Numerical Solutions of Stochastic Differential Equations Driven by Poisson Random Measure with Non-Lipschitz Coefficients

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Hui Yu ◽  
Minghui Song
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Linear Growth ◽  
Numerical Solutions ◽  
Random Measure ◽  
Euler Method ◽  
Poisson Random Measure ◽  
Euler’S Method ◽  
Linear Growth Condition

The numerical methods in the current known literature require the stochastic differential equations (SDEs) driven by Poisson random measure satisfying the global Lipschitz condition and the linear growth condition. In this paper, Euler's method is introduced for SDEs driven by Poisson random measure with non-Lipschitz coefficients which cover more classes of such equations than before. The main aim is to investigate the convergence of the Euler method in probability to such equations with non-Lipschitz coefficients. Numerical example is given to demonstrate our results.

Almost Sure Decay Stability of the Backward Euler-Maruyama Scheme for Stochastic Differential Equations with Unbounded Delay

2012 ◽  
Vol 235 ◽  
pp. 39-44 ◽  
Author(s):  
Lin Chen ◽  
Fu Ke Wu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Growth Condition ◽  
Linear Growth ◽  
Sufficient Conditions ◽  
Step Size ◽  
Unbounded Delay ◽  
Backward Euler ◽  
Linear Growth Condition ◽  
Highly Nonlinear

This paper deals with analytical and numerical stability properties of highly nonlinear stochastic differential equations (SDEs) with unbounded delay. Sufficient conditions for almost sure decay stability of previous system, almost sure decay stability of the backward Euler-Maruyama (BEM) methods are investigated. In \cite{Wu2010} and \cite{Mao2011}, the authors consider one-side linear growth condition and sufficient small step size. In this paper, we consider the monotone condition, which is weaker than one-side linear growth condition. And we only need a very weak restriction of the step size. Different from \cite{Szpruch2010}, Szpruch and Mao consider the asymptotic stability of the numerical approximate. In this paper we consider the almost sure decay stability of the numerical solution. This improves the existing results greatly.

scholarly journals Milstein-type semi-implicit split-step numerical methods for nonlinear stochastic differential equations with locally Lipschitz drift terms

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 1-12 ◽  
Author(s):  
Burhaneddin Izgi ◽  
Coskun Cetin
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Solutions ◽  
Rates Of Convergence ◽  
Ginzburg Landau ◽  
Linear Growth Condition ◽  
Locally Lipschitz ◽  
Non Linear ◽  
Ginzburg Landau Equations

We develop Milstein-type versions of semi-implicit split-step methods for numerical solutions of non-linear stochastic differential equations with locally Lipschitz coefficients. Under a one-sided linear growth condition on the drift term, we obtain some moment estimates and discuss convergence properties of these numerical methods. We compare the performance of multiple methods, including the backward Milstein, tamed Milstein, and truncated Milstein procedures on non-linear stochastic differential equations including generalized stochastic Ginzburg-Landau equations. In particular, we discuss their empirical rates of convergence.

scholarly journals Almost sure exponential stability of the θ-Euler-Maruyama method for neutral stochastic differential equations with time-dependent delay when θ ∈ [0; 1 2]

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5629-5645 ◽  
Author(s):  
Maja Obradovic ◽  
Marija Milosevic
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Stability Result ◽  
Time Dependent ◽  
Almost Sure Exponential Stability ◽  
Linear Growth Condition ◽  
Highly Nonlinear ◽  
The Stability ◽  
Drift Coefficient

This paper represents a generalization of the stability result on the Euler-Maruyama solution, which is established in the paper M. Milosevic, Almost sure exponential stability of solutions to highly nonlinear neutral stochastics differential equations with time-dependent delay and Euler-Maruyama approximation, Math. Comput. Model. 57 (2013) 887 - 899. The main aim of this paper is to reveal the sufficient conditions for the global almost sure asymptotic exponential stability of the ?-Euler-Maruyama solution (? ? [0, 1/2 ]), for a class of neutral stochastic differential equations with time-dependent delay. The existence and uniqueness of solution of the approximate equation is proved by employing the one-sided Lipschitz condition with respect to the both present state and delayed arguments of the drift coefficient of the equation. The technique used in proving the stability result required the assumption ? ?(0, 1/2], while the method is defined by employing the parameter ? with respect to the both drift coefficient and neutral term. Bearing in mind the difference between the technique which will be applied in the present paper and that used in the cited paper, the Euler-Maruyama case (? = 0) is considered separately. In both cases, the linear growth condition on the drift coefficient is applied, among other conditions. An example is provided to support the main result of the paper.

scholarly journals Stability of Nonlinear Neutral Stochastic Functional Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-26 ◽  
Author(s):  
Minggao Xue ◽  
Shaobo Zhou ◽  
Shigeng Hu
Keyword(s):  
Differential Equations ◽  
Growth Condition ◽  
Linear Growth ◽  
Functional Differential Equations ◽  
Stochastic Functional Differential Equations ◽  
Linear Growth Condition ◽  
Highly Nonlinear ◽  
Functional Differential ◽  
New Criteria ◽  
Stochastic Functional

Neutral stochastic functional differential equations (NSFDEs) have recently been studied intensively. The well-known conditions imposed for the existence and uniqueness and exponential stability of the global solution are the local Lipschitz condition and the linear growth condition. Therefore, the existing results cannot be applied to many important nonlinear NSFDEs. The main aim of this paper is to remove the linear growth condition and establish a Khasminskii-type test for nonlinear NSFDEs. New criteria not only cover a wide class of highly nonlinear NSFDEs but they can also be verified much more easily than the classical criteria. Finally, several examples are given to illustrate main results.

scholarly journals The Boundedness and Exponential Stability Criterions for Nonlinear Hybrid Neutral Stochastic Functional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Xiaofeng Zong ◽  
Fuke Wu ◽  
Chengming Huang
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Linear Growth ◽  
Functional Differential Equations ◽  
Stochastic Functional Differential Equations ◽  
Neutral Differential Equations ◽  
Linear Growth Condition ◽  
Functional Differential ◽  
Theoretical Results ◽  
Stochastic Functional

Neutral differential equations have been used to describe the systems that not only depend on the present and past states but also involve derivatives with delays. This paper considers hybrid nonlinear neutral stochastic functional differential equations (HNSFDEs) without the linear growth condition and examines the boundedness and exponential stability. Two illustrative examples are given to show the effectiveness of our theoretical results.

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