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Multi-step Richardson-Romberg Extrapolation: Remarks on Variance Control and Complexity

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  • Pagès Gilles

    (Email: gpa@ccr.jussieu.fr)

Abstract

We propose a multi-step Richardson-Romberg extrapolation method for the computation of expectations Ef(XT ) of a diffusion (Xt)t∈[0,T] when the weak time discretization error induced by the Euler scheme admits an expansion at an order R ≥ 2. The complexity of the estimator grows as R2 (instead of 2R in the classical method) and its variance is asymptotically controlled by considering some consistent Brownian increments in the underlying Euler schemes. Some Monte Carlo simulations were carried with path-dependent options (lookback, barrier) which support the conjecture that their weak time discretization error also admits an expansion (in a different scale). Then an appropriate Richardson-Romberg extrapolation seems to outperform the Euler scheme with Brownian bridge.

Suggested Citation

  • 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.
    • Handle: RePEc:bpj:mcmeap:v:13:y:2007:i:1:p:37-70:n:3
    DOI: 10.1515/MCMA.2007.003
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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. Arturo Kohatsu & Roger Pettersson, 2002. "Variance reduction methods for simulation of densities on Wiener space," Economics Working Papers 597, Department of Economics and Business, Universitat Pompeu Fabra.
    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. Bernard Lapeyre & Emmanuel Temam, 2001. "Competitive Monte Carlo methods for the pricing of Asian options," Post-Print hal-01667057, HAL.
    5. Guyon, Julien, 2006. "Euler scheme and tempered distributions," Stochastic Processes and their Applications, Elsevier, vol. 116(6), pages 877-904, June.
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    Cited by:

    1. Vincent Lemaire & Thibaut Montes & Gilles Pagès, 2020. "New Weak Error bounds and expansions for Optimal Quantization," Post-Print hal-02361644, HAL.
    2. Vincent Lemaire & Thibaut Montes & Gilles Pagès, 2019. "New Weak Error bounds and expansions for Optimal Quantization," Working Papers hal-02361644, HAL.

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