Nous étudions une équation différentielle stochastique de dimension dirigée par un processus de Lévy stable. Lorsque , nous examinons l’unicité trajectorielle pour cette équation. Quand , nous étudions une autre équation, équivalente en loi, mais pour laquelle l’unicité trajectorielle s’avère vraie sous des hypothèses bien plus faibles. Nous obtenons des résultats variés, selon que ou et selon que le processus stable dirigeant l’équation est symétrique ou non. Nos hypothèses concernent la régularité et la monotonie des coefficients de dérive et de diffusion.
We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order with drift and diffusion coefficients , . When , we investigate pathwise uniqueness for this equation. When , 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 or and on whether the driving stable process is symmetric or not. Our assumptions involve the regularity and monotonicity of and .
Mots-clés : stable processes, stochastic differential equations with jumps
@article{AIHPB_2013__49_1_138_0, author = {Fournier, Nicolas}, title = {On pathwise uniqueness for stochastic differential equations driven by stable {L\'evy} processes}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {138--159}, publisher = {Gauthier-Villars}, volume = {49}, number = {1}, year = {2013}, doi = {10.1214/11-AIHP420}, mrnumber = {3060151}, zbl = {1273.60069}, language = {en}, url = {http://archive.numdam.org/articles/10.1214/11-AIHP420/} }
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 SP - 138 EP - 159 VL - 49 IS - 1 PB - Gauthier-Villars UR - http://archive.numdam.org/articles/10.1214/11-AIHP420/ DO - 10.1214/11-AIHP420 LA - en ID - AIHPB_2013__49_1_138_0 ER -
%0 Journal Article %A Fournier, Nicolas %T On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes %J Annales de l'I.H.P. Probabilités et statistiques %D 2013 %P 138-159 %V 49 %N 1 %I Gauthier-Villars %U http://archive.numdam.org/articles/10.1214/11-AIHP420/ %R 10.1214/11-AIHP420 %G en %F AIHPB_2013__49_1_138_0
Fournier, Nicolas. On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 138-159. doi : 10.1214/11-AIHP420. http://archive.numdam.org/articles/10.1214/11-AIHP420/
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