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

Valentin Konakov; Stéphane Menozzi

doi:10.1007/s10959-010-0291-x

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

Valentin Konakov
, Stéphane Menozzi

Academy of Sciences
Laboratoire de Probabilités et Modèles Aléatoires

Research output: Contribution to journal › Article › peer-review

21 Citations (Scopus)

Abstract

Consider a multidimensional stochastic differential equation of the form X_t = x + ∫₀ ^t b(X_s-)ds + ∫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.

Original language	English
Pages (from-to)	454-478
Number of pages	25
Journal	Journal of Theoretical Probability
Volume	24
Issue number	2
DOIs	https://doi.org/10.1007/s10959-010-0291-x
Publication status	Published - 1 Jun 2011
Externally published	Yes

Keywords

Euler scheme
Parametrix
Symmetric stable processes

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10.1007/s10959-010-0291-x

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abstract = "Consider a multidimensional stochastic differential equation of the form Xt = x + ∫0 t b(Xs-)ds + ∫0 t f(Xs-)dZs, where (Zs)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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AB - Consider a multidimensional stochastic differential equation of the form Xt = x + ∫0 t b(Xs-)ds + ∫0 t f(Xs-)dZs, where (Zs)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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KW - Parametrix

KW - Symmetric stable processes

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ER -

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

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Keywords

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