On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes

Nicolas Fournier

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

  • Volume: 49, Issue: 1, page 138-159
  • ISSN: 0246-0203

Abstract

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We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order α with drift and diffusion coefficients b , σ . When α ∈ ( 1 , 2 ) , we investigate pathwise uniqueness for this equation. When α ∈ ( 0 , 1 ) , we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether α ∈ ( 0 , 1 ) or α ∈ ( 1 , 2 ) and on whether the driving stable process is symmetric or not. Our assumptions involve the regularity and monotonicity of b and σ .

How to cite

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Fournier, Nicolas. "On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes." Annales de l'I.H.P. Probabilités et statistiques 49.1 (2013): 138-159. <http://eudml.org/doc/272039>.

@article{Fournier2013,
abstract = {We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order $\alpha $ with drift and diffusion coefficients $b$, $\sigma $. When $\alpha \in (1,2)$, we investigate pathwise uniqueness for this equation. When $\alpha \in (0,1)$, we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether $\alpha \in (0,1)$ or $\alpha \in (1,2)$ and on whether the driving stable process is symmetric or not. Our assumptions involve the regularity and monotonicity of $b$ and $\sigma $.},
author = {Fournier, Nicolas},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stable processes; stochastic differential equations with jumps},
language = {eng},
number = {1},
pages = {138-159},
publisher = {Gauthier-Villars},
title = {On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes},
url = {http://eudml.org/doc/272039},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Fournier, Nicolas
TI - On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 1
SP - 138
EP - 159
AB - We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order $\alpha $ with drift and diffusion coefficients $b$, $\sigma $. When $\alpha \in (1,2)$, we investigate pathwise uniqueness for this equation. When $\alpha \in (0,1)$, we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether $\alpha \in (0,1)$ or $\alpha \in (1,2)$ and on whether the driving stable process is symmetric or not. Our assumptions involve the regularity and monotonicity of $b$ and $\sigma $.
LA - eng
KW - stable processes; stochastic differential equations with jumps
UR - http://eudml.org/doc/272039
ER -

References

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