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Adaptive weak approximation of reflected and stopped diffusions

Author

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  • Bayer Christian

    (Institute for Mathematics, Royal Institute of Technology, S-10044 Stockholm, Sweden, currently at Institute of Mathematics, TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany. E-mail: cbayer@kth.se)

  • Szepessy Anders

    (Institute for Mathematics, Royal Institute of Technology, S-10044 Stockholm, Sweden. E-mail: szepessy@kth.se)

  • Tempone Raúl

    (Applied Mathematics and Computational Sciences, KAUST, Thuwal, Saudi Arabia. E-mail: raul.tempone@kaust.edu.sa)

Abstract

We study the weak approximation problem of diffusions, which are reflected at a subset of the boundary of a domain and stopped at the remaining boundary. First, we derive an error representation for the projected Euler method of Costantini, Pacchiarotti and Sartoretto [Costantini et al., SIAM J. Appl. Math., 58(1):73–102, 1998], based on which we introduce two new algorithms. The first one uses a correction term from the representation in order to obtain a higher order of convergence, but the computation of the correction term is, in general, not feasible in dimensions d > 1. The second algorithm is adaptive in the sense of Moon, Szepessy, Tempone and Zouraris [Moon et al., Stoch. Anal. Appl., 23:511–558, 2005], using stochastic refinement of the time grid based on a computable error expansion derived from the representation. Regarding the stopped diffusion, it is based in the adaptive algorithm for purely stopped diffusions presented in Dzougoutov, Moon, von Schwerin, Szepessy and Tempone [Dzougoutov et al., Lect. Notes Comput. Sci. Eng., 44, 59–88, 2005]. We give numerical examples underlining the theoretical results.

Suggested Citation

  • Bayer Christian & Szepessy Anders & Tempone Raúl, 2010. "Adaptive weak approximation of reflected and stopped diffusions," Monte Carlo Methods and Applications, De Gruyter, vol. 16(1), pages 1-67, January.
    • Handle: RePEc:bpj:mcmeap:v:16:y:2010:i:1:p:1-67:n:1
    DOI: 10.1515/mcma.2010.001
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    References listed on IDEAS

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    1. Gobet, Emmanuel, 2000. "Weak approximation of killed diffusion using Euler schemes," Stochastic Processes and their Applications, Elsevier, vol. 87(2), pages 167-197, June.
    2. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
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    Cited by:

    1. Hajime Kawakami, 2015. "Reconstruction algorithm for unknown cavities via Feynman–Kac type formula," Computational Optimization and Applications, Springer, vol. 61(1), pages 101-133, May.

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