25 research outputs found
Mean-square convergence and stability of the backward Euler method for stochastic differential delay equations with highly nonlinear growing coefficients
Over the last few decades, the numerical methods for stochastic differential
delay equations (SDDEs) have been investigated and developed by many scholars.
Nevertheless, there is still little work to be completed. By virtue of the
novel technique, this paper focuses on the mean-square convergence and
stability of the backward Euler method (BEM) for SDDEs whose drift and
diffusion coefficients can both grow polynomially. The upper mean-square error
bounds of BEM are obtained. Then the convergence rate, which is one-half, is
revealed without using the moment boundedness of numerical solutions.
Furthermore, under fairly general conditions, the novel technique is applied to
prove that the BEM can inherit the exponential mean-square stability with a
simple proof. At last, two numerical experiments are implemented to illustrate
the reliability of the theories
Explicit Numerical Approximations for SDDEs in Finite and Infinite Horizons using the Adaptive EM Method: Strong Convergence and Almost Sure Exponential Stability
In this paper we investigate explicit numerical approximations for stochastic
differential delay equations (SDDEs) under a local Lipschitz condition by
employing the adaptive Euler-Maruyama (EM) method. Working in both finite and
infinite horizons, we achieve strong convergence results by showing the
boundedness of the pth moments of the adaptive EM solution. We also obtain the
order of convergence infinite horizon. In addition, we show almost sure
exponential stability of the adaptive approximate solution for both SDEs and
SDDEs