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Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme

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  • Gobet, Emmanuel
  • Menozzi, Stéphane

Abstract

We are interested in approximating a multidimensional hypoelliptic diffusion process (Xt)t[greater-or-equal, slanted]0 killed when it leaves a smooth domain D. When a discrete Euler scheme with time step h is used, we prove under a noncharacteristic boundary condition that the weak error is upper bounded by , generalizing the result obtained by Gobet in (Stoch. Proc. Appl. 87 (2000) 167) for the uniformly elliptic case. We also obtain a lower bound with the same rate , thus proving that the order of convergence is exactly 1/2. This provides a theoretical explanation of the well-known bias that we can numerically observe in that kind of procedure.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:112:y:2004:i:2:p:201-223
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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. Paolo Baldi & Lucia Caramellino & Maria Gabriella Iovino, 1999. "Pricing General Barrier Options: A Numerical Approach Using Sharp Large Deviations," Mathematical Finance, Wiley Blackwell, vol. 9(4), pages 293-321, October.
    3. BALLY Vlad & TALAY Denis, 1996. "The Law of the Euler Scheme for Stochastic Differential Equations: II. Convergence Rate of the Density," Monte Carlo Methods and Applications, De Gruyter, vol. 2(2), pages 93-128, December.
    4. Alsmeyer, Gerold, 1994. "On the Markov renewal theorem," Stochastic Processes and their Applications, Elsevier, vol. 50(1), pages 37-56, March.
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    Cited by:

    1. 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.
    2. 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.
    3. Remigijus Mikulevičius & Changyong Zhang, 2024. "Convergence of Weak Euler Approximation for Nondegenerate Stochastic Differential Equations Driven by Point and Martingale Measures," Journal of Theoretical Probability, Springer, vol. 37(1), pages 43-80, March.
    4. 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.
    5. Emmanuel Gobet, 2009. "Advanced Monte Carlo methods for barrier and related exotic options," Post-Print hal-00319947, HAL.
    6. 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.
    7. 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.
    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. repec:hal:wpaper:hal-00727430 is not listed on IDEAS

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