Weak antithetic MLMC estimation of SDEs with the Milstein scheme for low-dimensional Wiener processes

Kristian Debrabant*, Azadeh Ghasemifard, Nicky C. Mattsson

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

42 Downloads (Pure)

Abstract

In this paper, we implement a weak Milstein Scheme to simulate low-dimensional stochastic differential equations (SDEs). We prove that combining the antithetic multilevel Monte-Carlo (MLMC) estimator introduced by Giles and Szpruch with the MLMC approach for weak SDE approximation methods by Belomestny and Nagapetyan, we can achieve a quadratic computational complexity in the inverse of the Root Mean Square Error (RMSE) when estimating expected values of smooth functionals of SDE solutions, without simulating Lévy areas and without requiring any strong convergence of the underlying SDE approximation method. By using appropriate discrete variables this approach allows us to calculate the expectation on the coarsest level of resolution by enumeration, which, for low-dimensional problems, results in a reduced computational effort compared to standard MLMC sampling. These theoretical results are also confirmed by a numerical experiment.

Original languageEnglish
JournalApplied Mathematics Letters
Volume91
Pages (from-to)22-27
Number of pages6
ISSN0893-9659
DOIs
Publication statusPublished - May 2019

Fingerprint

Wiener Process
Stochastic Equations
Differential equations
Differential equation
Approximation Methods
Monte Carlo Sampling
Discrete Variables
Expected Value
Strong Convergence
Mean square error
Enumeration
Computational complexity
Computational Complexity
Numerical Experiment
Roots
Sampling
Estimator
Calculate
Experiments

Keywords

  • Milstein scheme
  • Multilevel Monte-Carlo
  • Stochastic differential equation
  • Weak approximation schemes

Cite this

@article{771b35dff99a4357b5d28a5489de03fa,
title = "Weak antithetic MLMC estimation of SDEs with the Milstein scheme for low-dimensional Wiener processes",
abstract = "In this paper, we implement a weak Milstein Scheme to simulate low-dimensional stochastic differential equations (SDEs). We prove that combining the antithetic multilevel Monte-Carlo (MLMC) estimator introduced by Giles and Szpruch with the MLMC approach for weak SDE approximation methods by Belomestny and Nagapetyan, we can achieve a quadratic computational complexity in the inverse of the Root Mean Square Error (RMSE) when estimating expected values of smooth functionals of SDE solutions, without simulating L{\'e}vy areas and without requiring any strong convergence of the underlying SDE approximation method. By using appropriate discrete variables this approach allows us to calculate the expectation on the coarsest level of resolution by enumeration, which, for low-dimensional problems, results in a reduced computational effort compared to standard MLMC sampling. These theoretical results are also confirmed by a numerical experiment.",
keywords = "Milstein scheme, Multilevel Monte-Carlo, Stochastic differential equation, Weak approximation schemes",
author = "Kristian Debrabant and Azadeh Ghasemifard and Mattsson, {Nicky C.}",
year = "2019",
month = "5",
doi = "10.1016/j.aml.2018.11.017",
language = "English",
volume = "91",
pages = "22--27",
journal = "Applied Mathematics Letters",
issn = "0893-9659",
publisher = "Pergamon Press",

}

Weak antithetic MLMC estimation of SDEs with the Milstein scheme for low-dimensional Wiener processes. / Debrabant, Kristian; Ghasemifard, Azadeh; Mattsson, Nicky C.

In: Applied Mathematics Letters, Vol. 91, 05.2019, p. 22-27.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Weak antithetic MLMC estimation of SDEs with the Milstein scheme for low-dimensional Wiener processes

AU - Debrabant, Kristian

AU - Ghasemifard, Azadeh

AU - Mattsson, Nicky C.

PY - 2019/5

Y1 - 2019/5

N2 - In this paper, we implement a weak Milstein Scheme to simulate low-dimensional stochastic differential equations (SDEs). We prove that combining the antithetic multilevel Monte-Carlo (MLMC) estimator introduced by Giles and Szpruch with the MLMC approach for weak SDE approximation methods by Belomestny and Nagapetyan, we can achieve a quadratic computational complexity in the inverse of the Root Mean Square Error (RMSE) when estimating expected values of smooth functionals of SDE solutions, without simulating Lévy areas and without requiring any strong convergence of the underlying SDE approximation method. By using appropriate discrete variables this approach allows us to calculate the expectation on the coarsest level of resolution by enumeration, which, for low-dimensional problems, results in a reduced computational effort compared to standard MLMC sampling. These theoretical results are also confirmed by a numerical experiment.

AB - In this paper, we implement a weak Milstein Scheme to simulate low-dimensional stochastic differential equations (SDEs). We prove that combining the antithetic multilevel Monte-Carlo (MLMC) estimator introduced by Giles and Szpruch with the MLMC approach for weak SDE approximation methods by Belomestny and Nagapetyan, we can achieve a quadratic computational complexity in the inverse of the Root Mean Square Error (RMSE) when estimating expected values of smooth functionals of SDE solutions, without simulating Lévy areas and without requiring any strong convergence of the underlying SDE approximation method. By using appropriate discrete variables this approach allows us to calculate the expectation on the coarsest level of resolution by enumeration, which, for low-dimensional problems, results in a reduced computational effort compared to standard MLMC sampling. These theoretical results are also confirmed by a numerical experiment.

KW - Milstein scheme

KW - Multilevel Monte-Carlo

KW - Stochastic differential equation

KW - Weak approximation schemes

U2 - 10.1016/j.aml.2018.11.017

DO - 10.1016/j.aml.2018.11.017

M3 - Journal article

AN - SCOPUS:85058209995

VL - 91

SP - 22

EP - 27

JO - Applied Mathematics Letters

JF - Applied Mathematics Letters

SN - 0893-9659

ER -