Reliability of difference analogues to preserve stability properties of stochastic Volterra integro-differential equations.

*(English)*Zbl 1133.65003Summary: We consider the reliability of some numerical methods in preserving the stability properties of the linear stochastic functional differential equation

\[ dx(t)=\biggl(\alpha x(t)+ \beta\int^t_0 x(s)\,ds\biggr)\, dt +\sigma x(t-\tau)\,dW(t), \] where \(\alpha,\beta,\sigma,\tau \geq 0\) are real constants, and \(W(t)\) is a standard Wiener process. The areas of the regions of asymptotic stability for the class of methods considered, indicated by the sufficient conditions for the discrete system, are shown to be equal in size to each other and we show that an upper bound can be put on the time-step parameter for the numerical method for which the system is asymptotically mean-square stable.

We illustrate our results by means of numerical experiments and various stability diagrams. We examine the extent to which the continuous system can tolerate stochastic perturbations before losing its stability properties and we illustrate how one may accurately choose a numerical method to preserve the stability properties of the original problem in the numerical solution. Our numerical experiments also indicate that the quality of the sufficient conditions is very high.

\[ dx(t)=\biggl(\alpha x(t)+ \beta\int^t_0 x(s)\,ds\biggr)\, dt +\sigma x(t-\tau)\,dW(t), \] where \(\alpha,\beta,\sigma,\tau \geq 0\) are real constants, and \(W(t)\) is a standard Wiener process. The areas of the regions of asymptotic stability for the class of methods considered, indicated by the sufficient conditions for the discrete system, are shown to be equal in size to each other and we show that an upper bound can be put on the time-step parameter for the numerical method for which the system is asymptotically mean-square stable.

We illustrate our results by means of numerical experiments and various stability diagrams. We examine the extent to which the continuous system can tolerate stochastic perturbations before losing its stability properties and we illustrate how one may accurately choose a numerical method to preserve the stability properties of the original problem in the numerical solution. Our numerical experiments also indicate that the quality of the sufficient conditions is very high.

##### MSC:

65C30 | Numerical solutions to stochastic differential and integral equations |

60H20 | Stochastic integral equations |

60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |

45J05 | Integro-ordinary differential equations |

45R05 | Random integral equations |

45M10 | Stability theory for integral equations |

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