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

Raphael Kruse; Yue Wu

doi:10.3934/dcdsb.2018253

Article Contents

2019, Volume 24, Issue 8: 3475-3502. Doi: 10.3934/dcdsb.2018253

This issue Previous Article Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises Next Article Numerical methods for PDE models related to pricing and expected lifetime of an extraction project under uncertainty

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

Raphael Kruse and
Yue Wu

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

Early access: August 23, 2018

Published online: August 23, 2018

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 65C30; Secondary: 65C05, 65L20, 60H10.

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Figure 1. Numerical experiment for SDE (48): Step sizes versus $L^2$ errors

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Figure 2. Numerical experiment for SDE (48): CPU time versus $L^2$ errors

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Table Listing 1. A sample implementation of (9) in Python

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