BIT Numerical Mathematics

, 49:113

Pathwise Taylor schemes for random ordinary differential equations

Article

DOI: 10.1007/s10543-009-0211-6

Cite this article as:
Jentzen, A. & Kloeden, P.E. Bit Numer Math (2009) 49: 113. doi:10.1007/s10543-009-0211-6

Abstract

Random ordinary differential equations (RODEs) are ordinary differential equations which contain a stochastic process in their vector fields. They can be analyzed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable. Traditional numerical schemes for ordinary differential equations thus do not achieve their usual order of convergence when applied to RODEs. Nevertheless, deterministic calculus can still be used to derive higher order numerical schemes for RODEs by means of a new kind of integral Taylor expansion. The theory is developed systematically here, applied to illustrative examples involving Brownian motion and fractional Brownian motion as the driving processes and compared with other numerical schemes for RODEs in the literature.

Keywords

Random ordinary differential equations Integral Taylor expansion One-step numerical scheme Pathwise convergence Brownian motion Fractional Brownian motion 

Mathematics Subject Classification (2000)

65C30 65L05 65L20 

Copyright information

© Springer Science + Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsJohann Wolfgang Goethe-UniversityFrankfurt am MainGermany

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