IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v120y2010i2p130-162.html
   My bibliography  Save this article

Stopped diffusion processes: Boundary corrections and overshoot

Author

Listed:
  • Gobet, Emmanuel
  • Menozzi, Stéphane

Abstract

For a stopped diffusion process in a multidimensional time-dependent domain , we propose and analyse a new procedure consisting in simulating the process with an Euler scheme with step size [Delta] and stopping it at discrete times in a modified domain, whose boundary has been appropriately shifted. The shift is locally in the direction of the inward normal n(t,x) at any point (t,x) on the parabolic boundary of , and its amplitude is equal to where [sigma] stands for the diffusion coefficient of the process. The procedure is thus extremely easy to use. In addition, we prove that the rate of convergence w.r.t. [Delta] for the associated weak error is higher than without shifting, generalizing the previous results by Broadie et al. (1997) [6] obtained for the one-dimensional Brownian motion. For this, we establish in full generality the asymptotics of the triplet exit time/exit position/overshoot for the discretely stopped Euler scheme. Here, the overshoot means the distance to the boundary of the process when it exits the domain. Numerical experiments support these results.

Suggested Citation

  • Gobet, Emmanuel & Menozzi, Stéphane, 2010. "Stopped diffusion processes: Boundary corrections and overshoot," Stochastic Processes and their Applications, Elsevier, vol. 120(2), pages 130-162, February.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:2:p:130-162
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(09)00165-3
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Rainer Avikainen, 2009. "On irregular functionals of SDEs and the Euler scheme," Finance and Stochastics, Springer, vol. 13(3), pages 381-401, September.
    2. Gobet, Emmanuel, 2000. "Weak approximation of killed diffusion using Euler schemes," Stochastic Processes and their Applications, Elsevier, vol. 87(2), pages 167-197, June.
    3. Mark Broadie & Paul Glasserman & Steven Kou, 1997. "A Continuity Correction for Discrete Barrier Options," Mathematical Finance, Wiley Blackwell, vol. 7(4), pages 325-349, October.
    4. Madalina Deaconu & Antoine Lejay, 2006. "A Random Walk on Rectangles Algorithm," Methodology and Computing in Applied Probability, Springer, vol. 8(1), pages 135-151, March.
    5. Gobet, Emmanuel & Menozzi, Stéphane, 2004. "Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 201-223, August.
    6. Alsmeyer, Gerold, 1994. "On the Markov renewal theorem," Stochastic Processes and their Applications, Elsevier, vol. 50(1), pages 37-56, March.
    7. Emmanuel Gobet, 2009. "Advanced Monte Carlo methods for barrier and related exotic options," Post-Print hal-00319947, HAL.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Cameron Martin & Hongyuan Zhang & Julia Costacurta & Mihai Nica & Adam R Stinchcombe, 2022. "Solving Elliptic Equations with Brownian Motion: Bias Reduction and Temporal Difference Learning," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1603-1626, September.
    2. Florian Bourgey & Emmanuel Gobet & Ying Jiao, 2022. "Bridging socioeconomic pathways of CO2 emission and credit risk," Post-Print hal-03458299, HAL.
    3. Herrmann, Samuel & Massin, Nicolas, 2023. "Exact simulation of the first passage time through a given level of jump diffusions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 553-576.
    4. Meier, Christian & Li, Lingfei & Zhang, Gongqiu, 2023. "Simulation of multidimensional diffusions with sticky boundaries via Markov chain approximation," European Journal of Operational Research, Elsevier, vol. 305(3), pages 1292-1308.
    5. Rey, Clément, 2019. "Approximation of Markov semigroups in total variation distance under an irregular setting: An application to the CIR process," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 539-571.
    6. Sheng-Feng Luo & Hsin-Chieh Wong, 2023. "Continuity correction: on the pricing of discrete double barrier options," Review of Derivatives Research, Springer, vol. 26(1), pages 51-90, April.
    7. repec:hal:wpaper:hal-03458299 is not listed on IDEAS
    8. Matoussi Anis & Sabbagh Wissal, 2016. "Numerical computation for backward doubly SDEs with random terminal time," Monte Carlo Methods and Applications, De Gruyter, vol. 22(3), pages 229-258, September.
    9. Rey Clément, 2017. "Convergence in total variation distance of a third order scheme for one-dimensional diffusion processes," Monte Carlo Methods and Applications, De Gruyter, vol. 23(1), pages 1-12, March.
    10. Christian Meier & Lingfei Li & Gongqiu Zhang, 2021. "Simulation of Multidimensional Diffusions with Sticky Boundaries via Markov Chain Approximation," Papers 2107.04260, arXiv.org.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Lejay, Antoine & Maire, Sylvain, 2007. "Computing the principal eigenvalue of the Laplace operator by a stochastic method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 73(6), pages 351-363.
    2. Mike Giles & Lukasz Szpruch, 2012. "Multilevel Monte Carlo methods for applications in finance," Papers 1212.1377, arXiv.org.
    3. Pagès Gilles, 2007. "Multi-step Richardson-Romberg Extrapolation: Remarks on Variance Control and Complexity," Monte Carlo Methods and Applications, De Gruyter, vol. 13(1), pages 37-70, April.
    4. Herrmann, Samuel & Massin, Nicolas, 2023. "Exact simulation of the first passage time through a given level of jump diffusions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 553-576.
    5. Emmanuel Gobet, 2009. "Advanced Monte Carlo methods for barrier and related exotic options," Post-Print hal-00319947, HAL.
    6. Aurélien Alfonsi & Benjamin Jourdain & Arturo Kohatsu-Higa, 2014. "Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme," Post-Print hal-00727430, HAL.
    7. Lucia Caramellino & Barbara Pacchiarotti & Simone Salvadei, 2015. "Large Deviation Approaches for the Numerical Computation of the Hitting Probability for Gaussian Processes," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 383-401, June.
    8. Cetin, Umut, 2018. "Diffusion transformations, Black-Scholes equation and optimal stopping," LSE Research Online Documents on Economics 87261, London School of Economics and Political Science, LSE Library.
    9. Caramellino Lucia & Pacchiarotti Barbara, 2002. "Sharp estimates for the hitting probability on time-dependent barriers for a Brownian Motion. Weak approximation of a Brownian motion killed on time-dependent barriers," Monte Carlo Methods and Applications, De Gruyter, vol. 8(3), pages 221-236, December.
    10. Hangsuck Lee & Gaeun Lee & Seongjoo Song, 2021. "Multi-step Reflection Principle and Barrier Options," Papers 2105.15008, arXiv.org.
    11. Hideyuki Tanaka & Toshihiro Yamada, 2012. "Strong Convergence for Euler-Maruyama and Milstein Schemes with Asymptotic Method," Papers 1210.0670, arXiv.org, revised Nov 2013.
    12. Xie, Fei & He, Zhijian & Wang, Xiaoqun, 2019. "An importance sampling-based smoothing approach for quasi-Monte Carlo simulation of discrete barrier options," European Journal of Operational Research, Elsevier, vol. 274(2), pages 759-772.
    13. N. Hilber & N. Reich & C. Schwab & C. Winter, 2009. "Numerical methods for Lévy processes," Finance and Stochastics, Springer, vol. 13(4), pages 471-500, September.
    14. F Bourgey & S de Marco & Emmanuel Gobet & Alexandre Zhou, 2020. "Multilevel Monte-Carlo methods and lower-upper bounds in Initial Margin computations," Post-Print hal-02430430, HAL.
    15. Casella, Bruno & Roberts, Gareth O., 2011. "Exact Simulation of Jump-Diffusion Processes with Monte Carlo Applications," MPRA Paper 95217, University Library of Munich, Germany.
    16. Hoel Håkon & von Schwerin Erik & Szepessy Anders & Tempone Raúl, 2014. "Implementation and analysis of an adaptive multilevel Monte Carlo algorithm," Monte Carlo Methods and Applications, De Gruyter, vol. 20(1), pages 1-41, March.
    17. Diana Dorobantu & Yahia Salhi & Pierre-Emmanuel Thérond, 2018. "Modelling net carrying amount of shares for market consistent valuation of life insurance liabilities," Working Papers hal-01840057, HAL.
    18. F Bourgey & S de Marco & Emmanuel Gobet & Alexandre Zhou, 2020. "Multilevel Monte-Carlo methods and lower-upper bounds in Initial Margin computations," Working Papers hal-02430430, HAL.
    19. Kontosakos, Vasileios E. & Mendonca, Keegan & Pantelous, Athanasios A. & Zuev, Konstantin M., 2021. "Pricing discretely-monitored double barrier options with small probabilities of execution," European Journal of Operational Research, Elsevier, vol. 290(1), pages 313-330.
    20. Jorge Ignacio Gonz'alez C'azares & Aleksandar Mijatovi'c & Ger'onimo Uribe Bravo, 2018. "Geometrically Convergent Simulation of the Extrema of L\'{e}vy Processes," Papers 1810.11039, arXiv.org, revised Jun 2021.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:120:y:2010:i:2:p:130-162. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.