The stochastic \(\Theta\)-method for nonlinear stochastic Volterra integro-differential equations
type: Item
instance of: scholarly article
title: The stochastic \(\Theta\)-method for nonlinear stochastic Volterra integro-differential equations
zbMATH Open document ID: 1474.65496
DOI: 10.1155/2014/583930
author: Peng Hu
author: Chengming Huang
published in: Abstract and Applied Analysis
publication date: 2019-02-14T00:00:00Z
review text: Summary: The stochastic \(\Theta\)-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 stochastic \(\Theta\)-method is convergent of order \(1 / 2\) in 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 stochastic \(\Theta\)-method is mean-square asymptotically stable for every stepsize if \(1 / 2 \leq \theta \leq 1\) and when \(0 \leq \theta < 1 / 2\), the stochastic \(\Theta\)-method is mean-square asymptotically stable for some small stepsizes. Finally, we validate our conclusions by numerical experiments.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45R05
zbMATH DE Number: 7022655
Wikidata QID: Q59039641
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/583930
OpenAlex ID: W2035945607
cites work: Volterra integral and differential equations
cites work: Collocation Methods for Volterra Integral and Related Functional Differential Equations
cites work: Numerical Methods for Nonlinear Volterra Integro-Differential Equations
cites work: Linear Multistep Methods for Volterra Integro-Differential Equations
cites work: Stability of stochastic integro differiential equations
cites work: Mean square stability of stochastic Volterra integro-differential equations
cites work: \(p\)th mean integrability and almost sure asymptotic stability of solutions of Itô-Volterra equations
cites work: Multistep methods for SDEs and their application to problems with small noise
cites work: Improved linear multi-step methods for stochastic ordinary differential equations
cites work: Numerical methods for strong solutions of stochastic differential equations: an overview
cites work: Implicit Taylor methods for stiff stochastic differential equations
cites work: Higher-order implicit strong numerical schemes for stochastic differential equations
cites work: Sample path approximation for stochastic integro-differential equations
cites work: Strong approximations of stochastic integro-differential equations
cites work: Reliability of difference analogues to preserve stability properties of stochastic Volterra integro-differential equations
cites work: Mean-Square and Asymptotic Stability of the Stochastic Theta Method
cites work: \(A\)-stability and stochastic mean-square stability
cites work: Continuous \(\Theta\)-methods for the stochastic pantograph equation
description: scientific article; zbMATH DE number 7022655