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On the Convergence Rates of IPA and FDC Derivative Estimators

Author

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  • Pierre L'Ecuyer

    (University of Montreal, Montreal, Canada)

  • Gaétan Perron

    (GESPRO Informatique, Ste. Foy, Quebec, Canada)

Abstract

We show that under the (sufficient) conditions usually given for infinitesimal perturbation analysis (IPA) to apply for derivative estimation, a finite-difference scheme with common random numbers (FDC) has the same order of convergence, namely O ( n −1/2 ), provided that the size of the finite-difference interval converges to zero fast enough. This holds for both one- and two-sided FDC. This also holds for different variants of IPA, such as some versions of smoothed perturbation analysis (SPA), which is based on conditional expectation. Finally, this also holds for the estimation of steady-state performance measures by truncated-horizon estimators, under some ergodicity assumptions. Our developments do not involve monotonicity, but are based on continuity and smoothness. We give examples and numerical illustrations which show that the actual difference in mean square error (MSE) between IPA and FDC is typically negligible. We also obtain the order of convergence of that difference, which is faster than the convergence of the MSE to zero.

Suggested Citation

  • Pierre L'Ecuyer & Gaétan Perron, 1994. "On the Convergence Rates of IPA and FDC Derivative Estimators," Operations Research, INFORMS, vol. 42(4), pages 643-656, August.
    • Handle: RePEc:inm:oropre:v:42:y:1994:i:4:p:643-656
    DOI: 10.1287/opre.42.4.643
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    Cited by:

    1. Hendrik Kohrs & Hermann Mühlichen & Benjamin R. Auer & Frank Schuhmacher, 2019. "Pricing and risk of swing contracts in natural gas markets," Review of Derivatives Research, Springer, vol. 22(1), pages 77-167, April.
    2. Sidney Yakowitz & Pierre L'Ecuyer & Felisa Vázquez-Abad, 2000. "Global Stochastic Optimization with Low-Dispersion Point Sets," Operations Research, INFORMS, vol. 48(6), pages 939-950, December.
    3. Montero, Miquel & Kohatsu-Higa, Arturo, 2003. "Malliavin Calculus applied to finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 320(C), pages 548-570.
    4. Boyle, Phelim & Broadie, Mark & Glasserman, Paul, 1997. "Monte Carlo methods for security pricing," Journal of Economic Dynamics and Control, Elsevier, vol. 21(8-9), pages 1267-1321, June.
    5. R. C. M. Brekelmans & L. T. Driessen & H. J. M. Hamers & D. Hertog, 2008. "Gradient Estimation Using Lagrange Interpolation Polynomials," Journal of Optimization Theory and Applications, Springer, vol. 136(3), pages 341-357, March.
    6. Pierre L’Ecuyer & Florian Puchhammer & Amal Ben Abdellah, 2022. "Monte Carlo and Quasi–Monte Carlo Density Estimation via Conditioning," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1729-1748, May.
    7. Benhamou, Eric, 2000. "A generalisation of Malliavin weighted scheme for fast computation of the Greeks," LSE Research Online Documents on Economics 119105, London School of Economics and Political Science, LSE Library.

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