Mean-square stability and convergence of a split-step theta method for stochastic Volterra integral equations

2021 ◽  
Vol 382 ◽  
pp. 113077
Author(s):  
Min Li ◽  
Chengming Huang ◽  
Peng Hu ◽  
Jiao Wen
Keyword(s):  
Integral Equations ◽  
Volterra Integral Equations ◽  
Stability And Convergence ◽  
Mean Square ◽  
Mean Square Stability ◽  
Theta Method

scholarly journals Mean-square stability of a class of stochastic integral equations

1972 ◽  
Vol 7 (3) ◽  
pp. 337-352
Author(s):  
W.J. Padgett
Keyword(s):  
Stochastic Process ◽  
Integral Equations ◽  
Stochastic Integral ◽  
Second Order ◽  
Mean Square ◽  
Random Solution ◽  
Mean Square Stability ◽  
Stochastic Integral Equations ◽  
Exponentially Stable ◽  
Image Position

The object of this paper is to investigate under very general conditions the existence and mean-square stability of a random solution of a class of stochastic integral equations in the formfor t ≥ 0, where a random solution is a second order stochastic process {x(t; w) t ≥ 0} which satisfies the equation almost certainly. A random solution x(t; w) is defined to be stable in mean-square if E[|x(t; w)|2] ≤ p for all t ≥ 0 and some p > 0 or exponentially stable in mean-square if E[|x(t; w)|2] ≤ pe-at, t ≥ 0, for some constants ρ > 0 and α > 0.

scholarly journals Numerical Study on Stochastic Diabetes Mellitus Model with Additive Noise

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Zhifang Zhang ◽  
Qingyi Zhan ◽  
Xiangdong Xie
Keyword(s):  
Diabetes Mellitus ◽  
Existence And Uniqueness ◽  
Additive Noise ◽  
Numerical Solutions ◽  
Numerical Study ◽  
Stability And Convergence ◽  
Mean Square ◽  
Existence And Uniqueness Theorem ◽  
Mean Square Stability ◽  
The Mean

This article focuses on the numerical analysis and simulation of the stochastic diabetes mellitus model with additive noise. The existence and uniqueness theorem of the solution under some appropriate assumptions is established. And, the mean square stability and convergence of numerical solutions are proposed, too. The practical use of these theorems is demonstrated in the numerical computations of the stochastic diabetes mellitus model and the value for the forecast of the tendency of diabetes mellitus in a given time.

scholarly journals Stability and Convergence of Solutions to Volterra Integral Equations on Time Scales

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Eleonora Messina ◽  
Antonia Vecchio
Keyword(s):  
Integral Equations ◽  
Time Scales ◽  
Sufficient Conditions ◽  
Volterra Integral Equations ◽  
Stability And Convergence ◽  
Time Behavior ◽  
Convergence Properties ◽  
Convergence Of Solutions ◽  
Long Time ◽  
The Stability

We consider Volterra integral equations on time scales and present our study about the long time behavior of their solutions. We provide sufficient conditions for the stability and investigate the convergence properties when the kernel of the equations vanishes at infinity.

scholarly journals Stability Issues for Selected Stochastic Evolutionary Problems: A Review

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 91 ◽  
Author(s):  
Angelamaria Cardone ◽  
Dajana Conte ◽  
Raffaele D’Ambrosio ◽  
Beatrice Paternoster
Keyword(s):  
Asymptotic Stability ◽  
Integral Equations ◽  
Numerical Approximation ◽  
Volterra Integral Equations ◽  
Mean Square ◽  
Stochastic Problems ◽  
Stability Issues ◽  
Stochastic Oscillators ◽  
Damped Oscillators

We review some recent contributions of the authors regarding the numerical approximation of stochastic problems, mostly based on stochastic differential equations modeling random damped oscillators and stochastic Volterra integral equations. The paper focuses on the analysis of selected stability issues, i.e., the preservation of the long-term character of stochastic oscillators over discretized dynamics and the analysis of mean-square and asymptotic stability properties of ϑ -methods for Volterra integral equations.

scholarly journals On Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Haiyan Yuan ◽  
Jihong Shen ◽  
Cheng Song
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Mean Square ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Mean Square Stability ◽  
Linear Growth Condition ◽  
The Mean ◽  
Delay Differential ◽  
Theta Method

A split-step theta (SST) method is introduced and used to solve the nonlinear neutral stochastic delay differential equations (NSDDEs). The mean square asymptotic stability of the split-step theta (SST) method for nonlinear neutral stochastic delay differential equations is studied. It is proved that under the one-sided Lipschitz condition and the linear growth condition, the split-step theta method withθ∈(1/2,1]is asymptotically mean square stable for all positive step sizes, and the split-step theta method withθ∈[0,1/2]is asymptotically mean square stable for some step sizes. It is also proved in this paper that the split-step theta (SST) method possesses a bounded absorbing set which is independent of initial data, and the mean square dissipativity of this method is also proved.

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