The Strong Convergence and Numerical Stability of Multistep Approximations of Solutions of Stochastic Ordinary Differential Equations

Abstract

Stochastic ordinary differential equations (SODEs) model physical phenomena driven by stochastic processes. As for deterministic differential equations, various numerical schemes have been proposed for SODEs. One of the key statements says that stability and consistency imply convergence of approximations to the exact solution for well-posed initial value problems.

In this work, we state and prove, correspondingly, generalized stochastic versions of the forementioned approximation principle for multistep methods. Theorems on numerical stability in the mean-square sense are proven, too.

Keywords:

• Mean-square convergence
• Multistep schemes
• Numerical schemes
• Stochastic ordinary differential equations

Mathematics Subject Classification:

• 60H10
• 65C05
• 65C20
• 65U05

Acknowledgments

Dedicated to ISTVÁN GYŐRI on the occasion of his sixtieth birthday.

The author gratefully acknowledges the support of the editor. She is thankful to the referee for careful reading and useful suggestions. The work of the author was supported by Hungarian National Scientific Grant (OTKA) T046929 and by Ministry of Science and Technology, Croatia Grant 00370114.