Convergence and stability of numerical methods with variable step size for stochastic pantograph differential equations

Abstract

The stochastic pantograph equations (SPEs) are very special stochastic delay differential equations (SDDEs) with unbounded memory. When the numerical methods with a constant step size are applied to the pantograph equations, the most difficult problem is the limited computer memory. In this paper, we construct methods with variable step size to solve SPEs. The analysis is motivated by the example of a mean-square stable linear SPE for which the Euler–Maruyama (EM) method with variable step size fails to reproduce this behaviour for any nonzero timestep. Then we consider the Backward Euler (BE) method with variable step size and develop the fundamental numerical analysis concerning its strong convergence and mean-square linear stability. It is proved that the numerical solutions produced by the BE method with variable step size converge to the exact solution under the local Lipschitz condition and the Bounded condition. Furthermore, the order of convergence p=½ is given under the Lipschitz condition. The result of the mean-square linear stability is given. Some illustrative numerical examples are presented to demonstrate the order of strong convergence and the mean-square linear stability of the BE method.

Keywords:

• stochastic pantograph equation
• mean-square stability
• Euler–Maruyama method
• Backward Euler method
• variable step size

2000 AMS Subject Classifications :

• 60H10
• 65C20
• 65C30

Acknowledgements

The authors would like to thank the anonymous referees and the associated editor for their helpful comments and suggestions.