# American Institute of Mathematical Sciences

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

show all references

##### 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
 [1] Wei Ouyang, Li Li. Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods. Journal of Industrial & Management Optimization, 2021, 17 (1) : 169-184. doi: 10.3934/jimo.2019105 [2] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [3] Baoli Yin, Yang Liu, Hong Li, Zhimin Zhang. Approximation methods for the distributed order calculus using the convolution quadrature. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1447-1468. doi: 10.3934/dcdsb.2020168 [4] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047 [5] Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324 [6] Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020367 [7] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468 [8] Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133 [9] José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091 [10] Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115 [11] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [12] Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391 [13] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050 [14] Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020126 [15] Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323 [16] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [17] Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control & Related Fields, 2021, 11 (1) : 73-94. doi: 10.3934/mcrf.2020028 [18] Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321 [19] John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044 [20] Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042

2019 Impact Factor: 1.27