scholarly journals The StochasticΘ-Method for Nonlinear Stochastic Volterra Integro-Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Peng Hu ◽  
Chengming Huang
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Numerical Experiments ◽  
Sufficient Condition ◽  
Asymptotically Stable ◽  
Mean Square ◽  
True Solution ◽  
Mean Square Exponential Stability ◽  
Mean Square Convergence ◽  
The Mean

The stochasticΘ-method is extended to solve nonlinear stochastic Volterra integro-differential equations. The mean-square convergence and asymptotic stability of the method are studied. First, we prove that the stochasticΘ-method is convergent of order1/2in mean-square sense for such equations. Then, a sufficient condition for mean-square exponential stability of the true solution is given. Under this condition, it is shown that the stochasticΘ-method is mean-square asymptotically stable for every stepsize if1/2≤θ≤1and when0≤θ<1/2, the stochasticΘ-method is mean-square asymptotically stable for some small stepsizes. Finally, we validate our conclusions by numerical experiments.

Strong Convergence Analysis of Split-Step θ-Scheme for Nonlinear Stochastic Differential Equations with Jumps

2016 ◽  
Vol 8 (6) ◽  
pp. 1004-1022 ◽  
Author(s):  
Xu Yang ◽  
Weidong Zhao
Keyword(s):  
Differential Equations ◽  
Theoretical Result ◽  
Strong Convergence ◽  
Stochastic Differential Equations ◽  
Rate Of Convergence ◽  
Numerical Experiments ◽  
Mean Square ◽  
Nonlinear Stochastic Differential Equations ◽  
Mean Square Convergence ◽  
The Mean

AbstractIn this paper, we investigate the mean-square convergence of the split-step θ-scheme for nonlinear stochastic differential equations with jumps. Under some standard assumptions, we rigorously prove that the strong rate of convergence of the split-step θ-scheme in strong sense is one half. Some numerical experiments are carried out to assert our theoretical result.

scholarly journals Combined Convex Technique on Delay-Distribution-Dependent Stability for Delayed Neural Networks

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Ting Wang ◽  
Tao Li ◽  
Mingxiang Xue ◽  
Shumin Fei
Keyword(s):  
Neural Networks ◽  
Functional Approach ◽  
Sufficient Condition ◽  
Asymptotically Stable ◽  
Mean Square ◽  
Variable Delay ◽  
Delayed Neural Networks ◽  
The Mean ◽  
Delay Partitioning ◽  
Delay Variation

Together with the Lyapunov-Krasovskii functional approach and an improved delay-partitioning idea, one novel sufficient condition is derived to guarantee a class of delayed neural networks to be asymptotically stable in the mean-square sense, in which the probabilistic variable delay and both of delay variation limits can be measured. Through combining the reciprocal convex technique and convex technique one, the criterion is presented via LMIs and its solvability heavily depends on the sizes of both time-delay range and its variations, which can become much less conservative than those present ones by thinning the delay intervals. Finally, it can be demonstrated by four numerical examples that our idea reduces the conservatism more effectively than some earlier reported ones.

scholarly journals Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion

2011 ◽  
Vol 2011 ◽  
pp. 1-22 ◽  
Author(s):  
Lin Hu ◽  
Siqing Gan
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Mean Square ◽  
Stochastic Delay Differential Equations ◽  
Implicit Methods ◽  
Stochastic Delay ◽  
Mean Square Convergence ◽  
One Step ◽  
The Mean ◽  
Delay Differential

A class of drift-implicit one-step schemes are proposed for the neutral stochastic delay differential equations (NSDDEs) driven by Poisson processes. A general framework for mean-square convergence of the methods is provided. It is shown that under certain conditions global error estimates for a method can be inferred from estimates on its local error. The applicability of the mean-square convergence theory is illustrated by the stochastic θ-methods and the balanced implicit methods. It is derived from Theorem 3.1 that the order of the mean-square convergence of both of them for NSDDEs with jumps is 1/2. Numerical experiments illustrate the theoretical results. It is worth noting that the results of mean-square convergence of the stochastic θ-methods and the balanced implicit methods are also new.

Asymptotic behaviour of immigration-branching processes by general set of types. II: Supercritical branching part

1975 ◽  
Vol 7 (03) ◽  
pp. 468-494
Author(s):  
H. Hering
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
Transition Function ◽  
Mean Square ◽  
Second Moments ◽  
Mean Square Convergence ◽  
The Mean ◽  
Parameter Dependent ◽  
Dependent Probability

We construct an immigration-branching process from an inhomogeneous Poisson process, a parameter-dependent probability distribution of populations and a Markov branching process with homogeneous transition function. The set of types is arbitrary, and the parameter is allowed to be discrete or continuous. Assuming a supercritical branching part with primitive first moments and finite second moments, we prove propositions on the mean square convergence and the almost sure convergence of normalized averaging processes associated with the immigration-branching process.

scholarly journals Attractor and Boundedness of Switched Stochastic Cohen-Grossberg Neural Networks

2016 ◽  
Vol 2016 ◽  
pp. 1-19 ◽  
Author(s):  
Chuangxia Huang ◽  
Jie Cao ◽  
Peng Wang
Keyword(s):  
Neural Networks ◽  
Sufficient Conditions ◽  
Average Dwell Time ◽  
Functional Method ◽  
Linear Matrix ◽  
Mean Square ◽  
Ultimate Boundedness ◽  
Mean Square Exponential Stability ◽  
The Mean ◽  
Uniformly Ultimate Boundedness

We address the problem of stochastic attractor and boundedness of a class of switched Cohen-Grossberg neural networks (CGNN) with discrete and infinitely distributed delays. With the help of stochastic analysis technology, the Lyapunov-Krasovskii functional method, linear matrix inequalities technique (LMI), and the average dwell time approach (ADT), some novel sufficient conditions regarding the issues of mean-square uniformly ultimate boundedness, the existence of a stochastic attractor, and the mean-square exponential stability for the switched Cohen-Grossberg neural networks are established. Finally, illustrative examples and their simulations are provided to illustrate the effectiveness of the proposed results.

Mean-Square Exponential Stability of Stochastic Interval Cellular Neural Networks with Time-Varying Delay

2013 ◽  
Vol 760-762 ◽  
pp. 1742-1747
Author(s):  
Jin Fang Han
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Cellular Neural Networks ◽  
Time Varying ◽  
Mean Square ◽  
Razumikhin Theorem ◽  
Time Varying Delay ◽  
Mean Square Exponential Stability ◽  
The Mean ◽  
Varying Delay

This paper is concerned with the mean-square exponential stability analysis problem for a class of stochastic interval cellular neural networks with time-varying delay. By using the stochastic analysis approach, employing Lyapunov function and norm inequalities, several mean-square exponential stability criteria are established in terms of the formula and Razumikhin theorem to guarantee the stochastic interval delayed cellular neural networks to be mean-square exponential stable. Some recent results reported in the literatures are generalized. A kind of equivalent description for this stochastic interval cellular neural networks with time-varying delay is also given.

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