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Implementation and Analysis of an Adaptive Multilevel Monte Carlo AlgorithmPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2012 (English)Report (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

2012. , p. 57
##### Series

TRITA-NA, ISSN 0348-2952 ; 2012:6
##### Keywords [en]

computational finance; Monte Carlo; multi-level; adaptivity; weak approximation
##### National Category

Probability Theory and Statistics Computer Engineering
##### Identifiers

URN: urn:nbn:se:kth:diva-94108OAI: oai:DiVA.org:kth-94108DiVA, id: diva2:525328
#####

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##### Note

##### In thesis

This work generalizes a multilevel Monte Carlo (MLMC) method in-troduced in [7] for the approximation of expected values of functions depending on the solution to an Ito stochastic differential equation. The work [7] proposed and analyzed a forward Euler MLMC method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a standard, single level, forward Euler Monte Carlo method from O( TOL^(−3) ) to O( TOL^(−2) log( TOL^(−1))^2 ) for a meansquare error of size 2 . This work uses instead a hierarchy of adaptivelyrefined, non uniform, time discretizations, generated by an adaptive algo-rithm introduced in [20, 19, 5]. Given a prescribed accuracy TOL in theweak error, this adaptive algorithm generates time discretizations basedon a posteriori expansions of the weak error first developed in [24]. Atheoretical analysis gives results on the stopping, the accuracy, and thecomplexity of the resulting adaptive MLMC algorithm. In particular, it isshown that: the adaptive refinements stop after a finite number of steps;the probability of the error being smaller than TOL is under certain as-sumptions controlled by a given confidence parameter, asymptotically asTOL → 0; the complexity is essentially the expected for MLMC methods,but with better control of the constant factors. We also show that themultilevel estimator is asymptotically normal using the Lindeberg-FellerCentral Limit Theorem. These theoretical results are based on previouslydeveloped single level estimates, and results on Monte Carlo stoppingfrom [3]. Our numerical tests include cases, one with singular drift andone with stopped diffusion, where the complexity of uniform single levelmethod is O TOL−4 . In both these cases the results confirm the theoryby exhibiting savings in the computational cost to achieve an accuracy of O(TOL), from O( TOL^(−3) )for the adaptive single level algorithm toessentially O( TOL^(−2) log(TOL−1)^2 ) for the adaptive MLMC.

QC 20120508

Available from: 2012-05-08 Created: 2012-05-07 Last updated: 2018-01-12Bibliographically approved1. Complexity and Error Analysis of Numerical Methods for Wireless Channels, SDE, Random Variables and Quantum Mechanics$(function(){PrimeFaces.cw("OverlayPanel","overlay525531",{id:"formSmash:j_idt904:0:j_idt908",widgetVar:"overlay525531",target:"formSmash:j_idt904:0:parentLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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