Convergence and almost sure polynomial stability of the backward and forward–backward Euler methods for highly nonlinear pantograph stochastic differential equations

2018 ◽  
Vol 150 ◽  
pp. 25-48 ◽  
Author(s):  
Marija Milošević
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Polynomial Stability ◽  
Backward Euler ◽  
Highly Nonlinear ◽  
Euler Methods

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 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.

Sign in / Sign up

Export Citation Format

Share Document