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.

ON ESTIMATION OF TRANSIENT STOCHASTIC STABILITY OF LINEAR SYSTEMS

2010 ◽  
Vol 10 (03) ◽  
pp. 385-405
Author(s):  
SK. SAFIQUE AHMAD ◽  
SOUMYENDU RAHA
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Linear Systems ◽  
Stochastic Stability ◽  
Transient Stability ◽  
Stochastic Systems ◽  
Unstable Equilibrium ◽  
Normal System ◽  
Logarithmic Norm ◽  
Transient Stabilization

In this work, we propose a measure for estimating the transient stability/stabilization of stochastic systems modeled with linear Itô Stochastic Differential Equations. The measure leads to useful and tractable computation of the stochastic Brockett version of the transient stabilization problems. Some properties and bound estimates of the measure which we call the Stochastic Logarithmic Norm are also discussed. The usefulness of the Stochastic Logarithmic Norm is illustrated with examples of unstable equilibrium systems and a non-normal system.

scholarly journals Second-order stochastic differential equations: stability, dissipativity, periodicity. V. - A survey

2021 ◽  
pp. 11-31
Author(s):  
Магомет Мишаустович Шумафов
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Stability ◽  
Stochastic Systems ◽  
Sufficient Conditions ◽  
Second Order ◽  
New Developments ◽  
Lyapunov’S Second Method ◽  
Basic Facts ◽  
Conditions Of Stability

Данная статья является продолжением предыдущей и представляет собой пятую, заключительную, часть работы автора. В работе делается обзор результатов исследований, касающихся свойств устойчивости, диссипативности и существования периодических решений стохастических дифференциальных уравнений и систем второго порядка. Приводятся результаты исследований, развивающие теорию устойчивости стохастических дифференциальных уравнений на основе модифицированного второго метода Ляпунова. Работа состоит из пяти частей. В первых двух частях были приведены предварительные сведения из теории вероятностей и случайных процессов, включая построение стохастических интегралов Ито и Стратоновича. В третьей части работы приведены некоторые факты из теории стохастических дифференциальных уравнений. Сформулированы теоремы существования и единственности для стохастических систем. В четвертой части приведены определения и даны основные сведения из теории устойчивости стохастических дифференциальных уравнений Ито. Общие теоремы об устойчивости, диссипативности и периодичности решений рассматриваемых систем сформулированы в терминах существования функций Ляпунова. В настоящей, пятой, части работы даны эффективные достаточные условия устойчивости по вероятности и экспоненциальной устойчивости в среднем квадратическом решений стохастических дифференциальных уравнений и систем второго порядка. Также даны достаточные условия диссипативности и периодичности случайных процессов, определяемых нелинейными дифференциальными уравнениями второго порядка со случайными правыми частями. В качестве примера рассматривается гармонический осциллятор, возмущенный белым шумом. В последнем разделе настоящей статьи сделан краткий обзор работ по стохастической устойчивости, которые характеризуют текущее состояние теории. This paper is a continuation of the previous papers and presents the fifth final part of the author’s work. The paper surveys the results concerning stability, dissipativity and periodicity properties of the second-order stochastic differential equations and systems. Some new developments in the theory of stability of stochastic differential equations based on the use of the modifying Lyapunov’s second method are presented. The work consists of five parts. In the first two parts we have introduced mathematical preliminaries from probability theory and stochastic processes including the construction of Ito and Stratonovich stochastic integrals. In the third part, some facts from the theory of stochastic differential equations are presented. The existence and uniqueness theorems for stochastic systems are formulated. In the fourth part, definitions are provided and basic facts from the theory of stability of stochastic differential equations are given. The basic general Lyapunov-like theorems on stochastic stability, dissipativity and periodicity for solutions of systems considered are formulated in the terms of the existence of Lyapunov functions. Here in the present fifth part, effective sufficient conditions of stability in probability, exponential stability in mean square for the second-order stochastic differential equations and systems are given. Also we give sufficient conditions for dissipativity and periodicity of random processes defined by nonlinear second-order differential equations with random right-hand sides. As an example the harmonic oscillator disturbed by white noise is considered. In the final section of the present paper, we briefly review some new publications related to stochastic stability that characterizes the state - of - the - art of the theory.

scholarly journals Mean-Square Stability of Split-Step Theta Milstein Methods for Stochastic Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Mahmoud A. Eissa ◽  
Haiying Zhang ◽  
Yu Xiao
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Mean Square ◽  
Step Size ◽  
Mean Square Stability ◽  
Test Equation ◽  
The Mean ◽  
The Stability ◽  
And Diffusion

The fundamental analysis of numerical methods for stochastic differential equations (SDEs) has been improved by constructing new split-step numerical methods. In this paper, we are interested in studying the mean-square (MS) stability of the new general drifting split-step theta Milstein (DSSθM) methods for SDEs. First, we consider scalar linear SDEs. The stability function of the DSSθM methods is investigated. Furthermore, the stability regions of the DSSθM methods are compared with those of test equation, and it is proved that the methods with θ≥3/2 are stochastically A-stable. Second, the nonlinear stability of DSSθM methods is studied. Under a coupled condition on the drifting and diffusion coefficients, it is proved that the methods with θ>1/2 can preserve the MS stability of the SDEs with no restriction on the step-size. Finally, numerical examples are given to examine the accuracy of the proposed methods under the stability conditions in approximation of SDEs.

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 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 A Type of Time-Symmetric Stochastic System and Related Games

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 118
Author(s):  
Qingfeng Zhu ◽  
Yufeng Shi ◽  
Jiaqiang Wen ◽  
Hui Zhang
Keyword(s):  
Nash Equilibrium ◽  
Time Delay ◽  
Differential Equations ◽  
Equilibrium Point ◽  
Stochastic Differential Equations ◽  
Stochastic System ◽  
Stochastic Systems ◽  
Necessary Condition ◽  
Nash Equilibrium Point ◽  
Doubly Stochastic

This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward–backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differential systems with time delay. Therefore, a class of nonzero sum differential game for doubly stochastic systems with time delay is studied in this paper. A necessary condition for the open-loop Nash equilibrium point of the Pontriagin-type maximum principle are established, and a sufficient condition for the Nash equilibrium point is obtained. Furthermore, the above results are applied to the study of nonzero sum differential games for linear quadratic backward doubly stochastic systems with delay. Based on the solution of FBDSDEs, an explicit expression of Nash equilibrium points for such game problems is established.

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