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    Adaptive Multilevel Monte Carlo Simulation

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    Type
    Conference Paper
    Authors
    Hoel, Hakon
    von Schwerin, Erik cc
    Szepessy, A
    Tempone, Raul cc
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Applied Mathematics and Computational Science Program
    Applied Mathematics and Computational Science Program
    Stochastic Numerics Research Group
    Date
    2011-08-23
    Online Publication Date
    2011-08-23
    Print Publication Date
    2012
    Permanent link to this record
    http://hdl.handle.net/10754/575782
    
    Metadata
    Show full item record
    Abstract
    This work generalizes a multilevel forward Euler Monte Carlo method introduced in Michael B. Giles. (Michael Giles. Oper. Res. 56(3):607–617, 2008.) for the approximation of expected values depending on the solution to an Itô stochastic differential equation. The work (Michael Giles. Oper. Res. 56(3):607– 617, 2008.) proposed and analyzed a forward Euler multilevelMonte Carlo 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. This work introduces an adaptive hierarchy of non uniform time discretizations, generated by an adaptive algorithmintroduced in (AnnaDzougoutov et al. Raùl Tempone. Adaptive Monte Carlo algorithms for stopped diffusion. In Multiscale methods in science and engineering, volume 44 of Lect. Notes Comput. Sci. Eng., pages 59–88. Springer, Berlin, 2005; Kyoung-Sook Moon et al. Stoch. Anal. Appl. 23(3):511–558, 2005; Kyoung-Sook Moon et al. An adaptive algorithm for ordinary, stochastic and partial differential equations. In Recent advances in adaptive computation, volume 383 of Contemp. Math., pages 325–343. Amer. Math. Soc., Providence, RI, 2005.). This form of the adaptive algorithm generates stochastic, path dependent, time steps and is based on a posteriori error expansions first developed in (Anders Szepessy et al. Comm. Pure Appl. Math. 54(10):1169– 1214, 2001). Our numerical results for a stopped diffusion problem, exhibit savings in the computational cost to achieve an accuracy of ϑ(TOL),from(TOL−3), from using a single level version of the adaptive algorithm to ϑ(((TOL−1)log(TOL))2).
    Citation
    Hoel, H., von Schwerin, E., Szepessy, A., & Tempone, R. (2011). Adaptive Multilevel Monte Carlo Simulation. Lecture Notes in Computational Science and Engineering, 217–234. doi:10.1007/978-3-642-21943-6_10
    Publisher
    Springer Nature
    Journal
    Lecture Notes in Computational Science and Engineering
    Conference/Event name
    Proceedings of a Winter Workshop at the Banff International Research Station
    ISBN
    978-3-642-21942-9
    DOI
    10.1007/978-3-642-21943-6_10
    ae974a485f413a2113503eed53cd6c53
    10.1007/978-3-642-21943-6_10
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