Hybrid Adaptive Multilevel Monte Carlo Algorithm for Non-Smooth Observables of Itô Stochastic Differential Equations

Handle URI:
http://hdl.handle.net/10754/306490
Title:
Hybrid Adaptive Multilevel Monte Carlo Algorithm for Non-Smooth Observables of Itô Stochastic Differential Equations
Authors:
Rached, Nadhir B.
Abstract:
The Monte Carlo forward Euler method with uniform time stepping is the standard technique to compute an approximation of the expected payoff of a solution of an Itô SDE. For a given accuracy requirement TOL, the complexity of this technique for well behaved problems, that is the amount of computational work to solve the problem, is O(TOL-3). A new hybrid adaptive Monte Carlo forward Euler algorithm for SDEs with non-smooth coefficients and low regular observables is developed in this thesis. This adaptive method is based on the derivation of a new error expansion with computable leading-order terms. The basic idea of the new expansion is the use of a mixture of prior information to determine the weight functions and posterior information to compute the local error. In a number of numerical examples the superior efficiency of the hybrid adaptive algorithm over the standard uniform time stepping technique is verified. When a non-smooth binary payoff with either GBM or drift singularity type of SDEs is considered, the new adaptive method achieves the same complexity as the uniform discretization with smooth problems. Moreover, the new developed algorithm is extended to the MLMC forward Euler setting which reduces the complexity from O(TOL-3) to O(TOL-2(log(TOL))2). For the binary option case with the same type of Itô SDEs, the hybrid adaptive MLMC forward Euler recovers the standard multilevel computational cost O(TOL-2(log(TOL))2). When considering a higher order Milstein scheme, a similar complexity result was obtained by Giles using the uniform time stepping for one dimensional SDEs. The difficulty to extend Giles' Milstein MLMC method to the multidimensional case is an argument for the flexibility of our new constructed adaptive MLMC forward Euler method which can be easily adapted to this setting. Similarly, the expected complexity O(TOL-2(log(TOL))2) is reached for the multidimensional case and verified numerically.
Advisors:
Tempone, Raul F.
Committee Member:
Alouini, Mohamed-Slim ( 0000-0003-4827-1793 ) ; Scavino, Marco
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Program:
Applied Mathematics and Computational Science
Issue Date:
Dec-2013
Type:
Thesis
Appears in Collections:
Applied Mathematics and Computational Science Program; Theses; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.advisorTempone, Raul F.en
dc.contributor.authorRached, Nadhir B.en
dc.date.accessioned2013-12-09T07:42:25Z-
dc.date.available2013-12-09T07:42:25Z-
dc.date.issued2013-12en
dc.identifier.urihttp://hdl.handle.net/10754/306490en
dc.description.abstractThe Monte Carlo forward Euler method with uniform time stepping is the standard technique to compute an approximation of the expected payoff of a solution of an Itô SDE. For a given accuracy requirement TOL, the complexity of this technique for well behaved problems, that is the amount of computational work to solve the problem, is O(TOL-3). A new hybrid adaptive Monte Carlo forward Euler algorithm for SDEs with non-smooth coefficients and low regular observables is developed in this thesis. This adaptive method is based on the derivation of a new error expansion with computable leading-order terms. The basic idea of the new expansion is the use of a mixture of prior information to determine the weight functions and posterior information to compute the local error. In a number of numerical examples the superior efficiency of the hybrid adaptive algorithm over the standard uniform time stepping technique is verified. When a non-smooth binary payoff with either GBM or drift singularity type of SDEs is considered, the new adaptive method achieves the same complexity as the uniform discretization with smooth problems. Moreover, the new developed algorithm is extended to the MLMC forward Euler setting which reduces the complexity from O(TOL-3) to O(TOL-2(log(TOL))2). For the binary option case with the same type of Itô SDEs, the hybrid adaptive MLMC forward Euler recovers the standard multilevel computational cost O(TOL-2(log(TOL))2). When considering a higher order Milstein scheme, a similar complexity result was obtained by Giles using the uniform time stepping for one dimensional SDEs. The difficulty to extend Giles' Milstein MLMC method to the multidimensional case is an argument for the flexibility of our new constructed adaptive MLMC forward Euler method which can be easily adapted to this setting. Similarly, the expected complexity O(TOL-2(log(TOL))2) is reached for the multidimensional case and verified numerically.en
dc.language.isoenen
dc.subjectMultilevel Monte Carloen
dc.subjectnon-smooth observablesen
dc.subjectIto^ SDEen
dc.subjectHybrid Adaptivityen
dc.titleHybrid Adaptive Multilevel Monte Carlo Algorithm for Non-Smooth Observables of Itô Stochastic Differential Equationsen
dc.typeThesisen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
thesis.degree.grantorKing Abdullah University of Science and Technologyen_GB
dc.contributor.committeememberAlouini, Mohamed-Slimen
dc.contributor.committeememberScavino, Marcoen
thesis.degree.disciplineApplied Mathematics and Computational Scienceen
thesis.degree.nameMaster of Scienceen
dc.person.id121851en
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