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Euler scheme and tempered distributions

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  • Guyon, Julien
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    Abstract

    Given a smooth -valued diffusion starting at point x, we study how fast the Euler scheme with time step 1/n converges in law to the random variable . To be precise, we look for the class of test functions f for which the approximate expectation converges with speed 1/n to . When f is smooth with polynomially growing derivatives or, under a uniform hypoellipticity condition for X, when f is only measurable and bounded, it is known that there exists a constant C1f(x) such that If X is uniformly elliptic, we expand this result to the case when f is a tempered distribution. In such a case, (resp. ) has to be understood as (resp. ) where p(t,x,[dot operator]) (resp. pn(t,x,[dot operator])) is the density of (resp. ). In particular, (1) is valid when f is a measurable function with polynomial growth, a Dirac mass or any derivative of a Dirac mass. We even show that (1) remains valid when f is a measurable function with exponential growth. Actually our results are symmetric in the two space variables x and y of the transition density and we prove that for a function and an O(1/n2) remainder rn which are shown to have gaussian tails and whose dependence on t is made precise. We give applications to option pricing and hedging, proving numerical convergence rates for prices, deltas and gammas.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 116 (2006)
    Issue (Month): 6 (June)
    Pages: 877-904

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    Handle: RePEc:eee:spapps:v:116:y:2006:i:6:p:877-904

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    Related research

    Keywords: Stochastic differential equation Euler scheme Rate of convergence Tempered distributions;

    References

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    1. 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.
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    Cited by:
    1. Benjamin Jourdain & Mohamed Sbai, 2013. "High order discretization schemes for stochastic volatility models," Post-Print hal-00409861, HAL.
    2. Rainer Avikainen, 2009. "On irregular functionals of SDEs and the Euler scheme," Finance and Stochastics, Springer, vol. 13(3), pages 381-401, September.
    3. Aurélien Alfonsi & Benjamin Jourdain & Arturo Kohatsu-Higa, 2012. "Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme," Working Papers hal-00727430, HAL.
    4. Nicola Bruti-Liberati & Eckhard Platen, 2006. "Approximation of Jump Diffusions in Finance and Economics," Research Paper Series 176, Quantitative Finance Research Centre, University of Technology, Sydney.

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