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August  2019, 24(8): 3475-3502. doi: 10.3934/dcdsb.2018253

## A randomized Milstein method for stochastic differential equations with non-differentiable drift coefficients

 Technische Universität Berlin, Institut für Mathematik, Secr. MA 5-3, Straße des 17. Juni 136, DE-10623 Berlin, Germany

Received  September 2017 Revised  February 2018 Published  August 2019 Early access  August 2018

Fund Project: The authors are supported by the German Research Foundation through FOR 2402.

In this paper a drift-randomized Milstein method is introduced for the numerical solution of non-autonomous stochastic differential equations with non-differentiable drift coefficient functions. Compared to standard Milstein-type methods we obtain higher order convergence rates in the $L^p(Ω)$ and almost sure sense. An important ingredient in the error analysis are randomized quadrature rules for Hölder continuous stochastic processes. By this we avoid the use of standard arguments based on the Itō-Taylor expansion which are typically applied in error estimates of the classical Milstein method but require additional smoothness of the drift and diffusion coefficient functions. We also discuss the optimality of our convergence rates. Finally, the question of implementation is addressed in a numerical experiment.

Citation: Raphael Kruse, Yue Wu. A randomized Milstein method for stochastic differential equations with non-differentiable drift coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3475-3502. doi: 10.3934/dcdsb.2018253
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##### References:
Numerical experiment for SDE (48): Step sizes versus $L^2$ errors
Numerical experiment for SDE (48): CPU time versus $L^2$ errors
A sample implementation of (9) in Python
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