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Stopped diffusion processes: Boundary corrections and overshoot

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  • 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.

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  • 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
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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. 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.
    3. 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.
    4. Alsmeyer, Gerold, 1994. "On the Markov renewal theorem," Stochastic Processes and their Applications, Elsevier, vol. 50(1), pages 37-56, March.
    5. Emmanuel Gobet, 2009. "Advanced Monte Carlo methods for barrier and related exotic options," Post-Print hal-00319947, HAL.
    6. Rainer Avikainen, 2009. "On irregular functionals of SDEs and the Euler scheme," Finance and Stochastics, Springer, vol. 13(3), pages 381-401, September.
    7. 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.
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    Cited by:

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    2. 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.
    3. Christian Meier & Lingfei Li & Gongqiu Zhang, 2021. "Simulation of Multidimensional Diffusions with Sticky Boundaries via Markov Chain Approximation," Papers 2107.04260, arXiv.org.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.

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