scholarly journals Stability of a Class of Hybrid Neutral Stochastic Differential Equations with Unbounded Delay

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
Boliang Lu ◽  
Ruili Song
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Lyapunov Functions ◽  
Stability Criteria ◽  
Bounded Delay ◽  
Unbounded Delay ◽  
The Stability ◽  
Generalized Itô Formula ◽  
Stability Results

This paper studies the stability of hybrid neutral stochastic differential equations with unbounded delay. Some novel exponential stability criteria and boundedness conditions are established based on the generalized Itô formula and Lyapunov functions. The factor e-εδ(t) is used to overcome the difficulties caused by the unbounded delay δ(t) effectively. In particular, our results generalize and improve some previous stability results from bounded delay to unbounded delay conditions. Finally, an example is presented to demonstrate the effectiveness of the proposed results.

scholarly journals The Mean Stability Criteria in terms of Two Measures for Stochastic Differential Equations with Coefficient’s Uncertainty

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Rui Zhang ◽  
Yinjing Guo ◽  
Xiangrong Wang ◽  
Xueqing Zhang
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Stability ◽  
Stochastic Systems ◽  
Stability Criteria ◽  
Two Measures ◽  
The Mean ◽  
The Stability

This paper extends the stochastic stability criteria of two measures to the mean stability and proves the stability criteria for a kind of stochastic Itô’s systems. Moreover, by applying optimal control approaches, the mean stability criteria in terms of two measures are also obtained for the stochastic systems with coefficient’s uncertainty.

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 solutions of Caputo fractional stochastic differential equations

2021 ◽  
Vol 26 (4) ◽  
pp. 581-596
Author(s):  
Guanli Xiao ◽  
JinRong Wang
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Stochastic Stability ◽  
Stopping Time ◽  
Asymptotical Stability ◽  
Stability Of Solutions ◽  
Numerical Examples ◽  
Fractional Stochastic Differential Equations ◽  
The Stability

In this paper, we study the stability of Caputo-type fractional stochastic differential equations. Stochastic stability and stochastic asymptotical stability are shown by stopping time technique. Almost surly exponential stability and pth moment exponentially stability are derived by a new established Itô’s formula of Caputo version. Numerical examples are given to illustrate the main results.

scholarly journals Convergence and stability of the one-leg θ method for stochastic differential equations with piecewise continuous arguments

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 945-960
Author(s):  
Yulan Lu ◽  
Minghui Song ◽  
Mingzhu Liu
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Step Size ◽  
Almost Sure Exponential Stability ◽  
Exponentially Stable ◽  
Piecewise Continuous ◽  
Piecewise Continuous Arguments ◽  
The Stability ◽  
The One

The equivalent relation is established here about the stability of stochastic differential equations with piecewise continuous arguments(SDEPCAs) and that of the one-leg ? method applied to the SDEPCAs. Firstly, the convergence of the one-leg ? method to SDEPCAs under the global Lipschitz condition is proved. Secondly, it is proved that the SDEPCAs are pth(p 2 (0; 1)) moment exponentially stable if and only if the one-leg ? method is pth moment exponentially stable for some sufficiently small step-size. Thirdly, the corollaries that the pth moment exponential stability of the SDEPCAs (the one-leg ? method) implies the almost sure exponential stability of the SDEPCAs (the one-leg ? method) are given. Finally, numerical simulations are provided to illustrate the theoretical results.

scholarly journals Stabilization of Stochastic Differential Equations Driven by G-Brownian Motion with Aperiodically Intermittent Control

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 988
Author(s):  
Pengju Duan
Keyword(s):  
Brownian Motion ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Mild Solution ◽  
Intermittent Control

The paper is devoted to studying the exponential stability of a mild solution of stochastic differential equations driven by G-Brownian motion with an aperiodically intermittent control. The aperiodically intermittent control is added into the drift coefficients, when intermittent intervals and coefficients satisfy suitable conditions; by use of the G-Lyapunov function, the p-th exponential stability is obtained. Finally, an example is given to illustrate the availability of the obtained results.

Stability of stochastic differential equations driven by multifractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Oussama El Barrimi ◽  
Youssef Ouknine
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stationary Process ◽  
Strong Stability ◽  
Pathwise Uniqueness ◽  
Uniqueness Property ◽  
Selection Theorem ◽  
Multifractional Brownian Motion ◽  
Stability Results

Abstract Our aim in this paper is to establish some strong stability results for solutions of stochastic differential equations driven by a Riemann–Liouville multifractional Brownian motion. The latter is defined as a Gaussian non-stationary process with a Hurst parameter as a function of time. The results are obtained assuming that the pathwise uniqueness property holds and using Skorokhod’s selection theorem.

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