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Adaptive Multi Level Monte Carlo Simulation
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
Applied Mathematics and Computational Sciences, KAUST, Thuwal, Saudi Arabia.
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
2012 (English)In: Numerical Analysis of Multiscale Computations: Proceedings of a Winter Workshop at the Banff International Research Station 2009, Springer, 2012, Vol. 82, 217-234 p.Conference paper, Published paper (Refereed)
Abstract [en]

This work generalizes a multilevel Forward Euler Monte Carlo methodintroduced in [5] for the approximation of expected values depending onthe solution to an Itˆo stochastic differential equation. The work [5] proposedand analyzed a Forward Euler Multilevel Monte Carlo method basedon a hierarchy of uniform time discretizations and control variates to reducethe computational effort required by a standard, single level, ForwardEuler Monte Carlo method. This work introduces an adaptive hierarchyof non uniform time discretizations, generated by adaptive algorithms introducedin [11, 10]. These adaptive algorithms apply either deterministictime steps or stochastic time steps and are based on a posteriori error expansionsfirst developed in [14]. Under sufficient regularity conditions, ournumerical results, which include one case with singular drift and one withstopped diffusion, exhibit savings in the computational cost to achieve anaccuracy of O(TOL), from O`TOL−3´to O“`TOL−1 log (TOL)´2”. Wealso include an analysis of a simplified version of the adaptive algorithmfor which we prove similar accuracy and computational cost results.

Place, publisher, year, edition, pages
Springer, 2012. Vol. 82, 217-234 p.
Series
Lecture Notes in Computational Science and Engineering, ISSN 1439-7358 ; 82
Keyword [en]
computational finance, Monte Carlo, multi-level, adaptivity, weak approximation, error control, Euler–Maruyama method, a posteriori error estimates, backward dual functions, adjoints
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:kth:diva-12918DOI: 10.1007/978-3-642-21943-6_10ISBN: 978-3-642-21942-9 (print)OAI: oai:DiVA.org:kth-12918DiVA: diva2:319597
Conference
Winter Workshop at the Banff International Research Station 2009
Funder
Swedish e‐Science Research Center
Note

QC 20120124

Available from: 2010-05-18 Created: 2010-05-18 Last updated: 2017-01-17Bibliographically approved
In thesis
1. Coarse Graining Monte Carlo Methods for Wireless Channels and Stochastic Differential Equations
Open this publication in new window or tab >>Coarse Graining Monte Carlo Methods for Wireless Channels and Stochastic Differential Equations
2010 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of two papers considering different aspects of stochastic process modelling and the minimisation of computational cost.

In the first paper, we analyse statistical signal properties and develop a Gaussian pro- cess model for scenarios with a moving receiver in a scattering environment, as in Clarke’s model, with the generalisation that noise is introduced through scatterers randomly flip- ping on and off as a function of time. The Gaussian process model is developed by extracting mean and covariance properties from the Multipath Fading Channel model (MFC) through coarse graining. That is, we verify that under certain assumptions, signal realisations of the MFC model converge to a Gaussian process and thereafter compute the Gaussian process’ covariance matrix, which is needed to construct Gaussian process signal realisations. The obtained Gaussian process model is under certain assumptions less computationally costly, containing more channel information and having very similar signal properties to its corresponding MFC model. We also study the problem of fitting our model’s flip rate and scatterer density to measured signal data.

The second paper generalises a multilevel Forward Euler Monte Carlo method intro- duced by Giles [1] for the approximation of expected values depending on the solution to an Ito stochastic differential equation. Giles work [1] proposed and analysed a Forward Euler Multilevel Monte Carlo method based on realsiations on a hierarchy of uniform time discretisations and a coarse graining based control variates idea to reduce the computa- tional effort required by a standard single level Forward Euler Monte Carlo method. This work introduces an adaptive hierarchy of non uniform time discretisations generated by adaptive algorithms developed by Moon et al. [3, 2]. These adaptive algorithms apply either deterministic time steps or stochastic time steps and are based on a posteriori error expansions first developed by Szepessy et al. [4]. Under sufficient regularity conditions, our numerical results, which include one case with singular drift and one with stopped dif- fusion, exhibit savings in the computational cost to achieve an accuracy of O(T ol), from O(T ol−3 ) to O (log (T ol) /T ol)2 . We also include an analysis of a simplified version of the adaptive algorithm for which we prove similar accuracy and computational cost results.

 

Place, publisher, year, edition, pages
Stockholm: KTH, 2010. viii, 20 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2010:07
Keyword
Coarse graining, Monte Carlo Methods, Stochastic processes
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-12897 (URN)978-91-7415-686-7 (ISBN)
Presentation
2010-06-11, E2, Lindstedtsvägen 3, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Available from: 2010-05-18 Created: 2010-05-18 Last updated: 2010-05-19Bibliographically approved

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