# Pathwise Approximation of Random Ordinary Differential Equations

L. Grüne, P. E. Kloeden:
**Pathwise Approximation of Random Ordinary Differential Equations**

*BIT - Numerical Mathematics*
**41**,
710 - 721,
2001

DOI:
10.1023/A:1021995918864

Keywords: Euler method; averaging method; error reduction; Heun methods; random ordinary differential equation; convergence

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## Abstract:

Standard error estimates for one-step numerical schemes for nonautonomous ordinary differential equations usually assume appropriate smoothness in both time and state variables and thus are not suitable for the pathwise approximation of random ordinary differential equations which are typically at most continuous or Hölder continuous in the time variable. Here it is shown that the usual higher order of convergence can be retained if one first averages the time dependence over each discretization subinterval.