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Weak Error for Stable Driven Stochastic Differential Equations: Expansion of the Densities

Author

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  • Valentin Konakov

    (Academy of Sciences)

  • Stéphane Menozzi

    (Université Paris VII Diderot)

Abstract

Consider a multidimensional stochastic differential equation of the form $X_{t}=x+\int_{0}^{t}b(X_{s-})\,ds+\int_{0}^{t}f(X_{s-})\,dZ_{s}$ , where (Z s )s≥0 is a symmetric stable process. Under suitable assumptions on the coefficients, the unique strong solution of the above equation admits a density with respect to Lebesgue measure, and so does its Euler scheme. Using a parametrix approach, we derive an error expansion with respect to the time step for the difference of these densities.

Suggested Citation

  • Valentin Konakov & Stéphane Menozzi, 2011. "Weak Error for Stable Driven Stochastic Differential Equations: Expansion of the Densities," Journal of Theoretical Probability, Springer, vol. 24(2), pages 454-478, June.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:2:d:10.1007_s10959-010-0291-x
    DOI: 10.1007/s10959-010-0291-x
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    References listed on IDEAS

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    1. Guyon, Julien, 2006. "Euler scheme and tempered distributions," Stochastic Processes and their Applications, Elsevier, vol. 116(6), pages 877-904, June.
    2. Imkeller, P. & Pavlyukevich, I., 2006. "First exit times of SDEs driven by stable Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 116(4), pages 611-642, April.
    3. Aleksander Janicki & Zbigniew Michna & Aleksander Weron, 1996. "Approximation of stochastic differential equations driven by alpha-stable Levy motion," HSC Research Reports HSC/96/02, Hugo Steinhaus Center, Wroclaw University of Technology.
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