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Pathwise Taylor schemes for random ordinary differential equations

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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.

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Authors and Affiliations

  1. Department of Mathematics, Johann Wolfgang Goethe-University, 60054, Frankfurt am Main, Germany

    Arnulf Jentzen & Peter E. Kloeden

Authors
  1. Arnulf Jentzen
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  2. Peter E. Kloeden
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Corresponding author

Correspondence to Arnulf Jentzen.

Additional information

Communicated by Anders Szepessy.

Partially supported by the DFG project “Pathwise numerics and dynamics of stochastic evolution equations”.

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Jentzen, A., Kloeden, P.E. Pathwise Taylor schemes for random ordinary differential equations. Bit Numer Math 49, 113–140 (2009). https://doi.org/10.1007/s10543-009-0211-6

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  • DOI: https://doi.org/10.1007/s10543-009-0211-6

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