Stochastic differential equations with Sobolev drifts and driven by α -stable processes

Xicheng Zhang

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 4, page 1057-1079
  • ISSN: 0246-0203

Abstract

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In this article we prove the pathwise uniqueness for stochastic differential equations in ℝ d with time-dependent Sobolev drifts, and driven by symmetric α -stable processes provided that α ∈ ( 1 , 2 ) and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when α ∈ ( 2 d d + 1 , 2 ) . Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales.

How to cite

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Zhang, Xicheng. "Stochastic differential equations with Sobolev drifts and driven by $\alpha $-stable processes." Annales de l'I.H.P. Probabilités et statistiques 49.4 (2013): 1057-1079. <http://eudml.org/doc/272007>.

@article{Zhang2013,
abstract = {In this article we prove the pathwise uniqueness for stochastic differential equations in $\mathbb \{R\}^\{d\}$ with time-dependent Sobolev drifts, and driven by symmetric $\alpha $-stable processes provided that $\alpha \in (1,2)$ and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when $\alpha \in (\frac\{2d\}\{d+1\},2)$. Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales.},
author = {Zhang, Xicheng},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {pathwise uniqueness; symmetric $\alpha $-stable process; Krylov’s estimate; fractional Sobolev space; pathwise uniqueness; symmetric -stable process; Krylov's estimate; fractional Sobolev space},
language = {eng},
number = {4},
pages = {1057-1079},
publisher = {Gauthier-Villars},
title = {Stochastic differential equations with Sobolev drifts and driven by $\alpha $-stable processes},
url = {http://eudml.org/doc/272007},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Zhang, Xicheng
TI - Stochastic differential equations with Sobolev drifts and driven by $\alpha $-stable processes
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 4
SP - 1057
EP - 1079
AB - In this article we prove the pathwise uniqueness for stochastic differential equations in $\mathbb {R}^{d}$ with time-dependent Sobolev drifts, and driven by symmetric $\alpha $-stable processes provided that $\alpha \in (1,2)$ and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when $\alpha \in (\frac{2d}{d+1},2)$. Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales.
LA - eng
KW - pathwise uniqueness; symmetric $\alpha $-stable process; Krylov’s estimate; fractional Sobolev space; pathwise uniqueness; symmetric -stable process; Krylov's estimate; fractional Sobolev space
UR - http://eudml.org/doc/272007
ER -

References

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