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

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.

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Correspondence to Stéphane Menozzi.

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Konakov, V., Menozzi, S. Weak Error for Stable Driven Stochastic Differential Equations: Expansion of the Densities. J Theor Probab 24, 454–478 (2011). https://doi.org/10.1007/s10959-010-0291-x

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Keywords

  • Symmetric stable processes
  • Parametrix
  • Euler scheme

Mathematics Subject Classification (2000)

  • 60H30
  • 65C30
  • 60G52