TY - JOUR

T1 - Implementation and analysis of an adaptive multilevel Monte Carlo algorithm

AU - Hoel, Hakon

AU - Von Schwerin, Erik

AU - Szepessy, Anders

AU - Tempone, Raul

N1 - KAUST Repository Item: Exported on 2020-10-01

PY - 2014/1/1

Y1 - 2014/1/1

N2 - We present an adaptive multilevel Monte Carlo (MLMC) method for weak approximations of solutions to Itô stochastic dierential equations (SDE). The work [11] proposed and analyzed an MLMC method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a single level Euler-Maruyama Monte Carlo method from O(TOL-3) to O(TOL-2 log(TOL-1)2) for a mean square error of O(TOL2). Later, the work [17] presented an MLMC method using a hierarchy of adaptively re ned, non-uniform time discretizations, and, as such, it may be considered a generalization of the uniform time discretizationMLMC method. This work improves the adaptiveMLMC algorithms presented in [17] and it also provides mathematical analysis of the improved algorithms. In particular, we show that under some assumptions our adaptive MLMC algorithms are asymptotically accurate and essentially have the correct complexity but with improved control of the complexity constant factor in the asymptotic analysis. Numerical tests include one case with singular drift and one with stopped diusion, where the complexity of a uniform single level method is O(TOL-4). For both these cases the results con rm the theory, exhibiting savings in the computational cost for achieving the accuracy O(TOL) from O(TOL-3) for the adaptive single level algorithm to essentially O(TOL-2 log(TOL-1)2) for the adaptive MLMC algorithm. © 2014 by Walter de Gruyter Berlin/Boston 2014.

AB - We present an adaptive multilevel Monte Carlo (MLMC) method for weak approximations of solutions to Itô stochastic dierential equations (SDE). The work [11] proposed and analyzed an MLMC method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a single level Euler-Maruyama Monte Carlo method from O(TOL-3) to O(TOL-2 log(TOL-1)2) for a mean square error of O(TOL2). Later, the work [17] presented an MLMC method using a hierarchy of adaptively re ned, non-uniform time discretizations, and, as such, it may be considered a generalization of the uniform time discretizationMLMC method. This work improves the adaptiveMLMC algorithms presented in [17] and it also provides mathematical analysis of the improved algorithms. In particular, we show that under some assumptions our adaptive MLMC algorithms are asymptotically accurate and essentially have the correct complexity but with improved control of the complexity constant factor in the asymptotic analysis. Numerical tests include one case with singular drift and one with stopped diusion, where the complexity of a uniform single level method is O(TOL-4). For both these cases the results con rm the theory, exhibiting savings in the computational cost for achieving the accuracy O(TOL) from O(TOL-3) for the adaptive single level algorithm to essentially O(TOL-2 log(TOL-1)2) for the adaptive MLMC algorithm. © 2014 by Walter de Gruyter Berlin/Boston 2014.

UR - http://hdl.handle.net/10754/563323

UR - https://www.degruyter.com/view/j/mcma.2014.20.issue-1/mcma-2013-0014/mcma-2013-0014.xml

UR - http://www.scopus.com/inward/record.url?scp=84896789808&partnerID=8YFLogxK

U2 - 10.1515/mcma-2013-0014

DO - 10.1515/mcma-2013-0014

M3 - Article

VL - 20

SP - 1

EP - 41

JO - Monte Carlo Methods and Applications

JF - Monte Carlo Methods and Applications

SN - 0929-9629

IS - 1

ER -