Strong solutions for stochastic differential equations with jumps

Zenghu Li; Leonid Mytnik

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 4, page 1055-1067
  • ISSN: 0246-0203

Abstract

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General stochastic equations with jumps are studied. We provide criteria for the uniqueness and existence of strong solutions under non-Lipschitz conditions of Yamada–Watanabe type. The results are applied to stochastic equations driven by spectrally positive Lévy processes.

How to cite

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Li, Zenghu, and Mytnik, Leonid. "Strong solutions for stochastic differential equations with jumps." Annales de l'I.H.P. Probabilités et statistiques 47.4 (2011): 1055-1067. <http://eudml.org/doc/243441>.

@article{Li2011,
abstract = {General stochastic equations with jumps are studied. We provide criteria for the uniqueness and existence of strong solutions under non-Lipschitz conditions of Yamada–Watanabe type. The results are applied to stochastic equations driven by spectrally positive Lévy processes.},
author = {Li, Zenghu, Mytnik, Leonid},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stochastic equation; strong solution; pathwise uniqueness; non-Lipschitz condition},
language = {eng},
number = {4},
pages = {1055-1067},
publisher = {Gauthier-Villars},
title = {Strong solutions for stochastic differential equations with jumps},
url = {http://eudml.org/doc/243441},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Li, Zenghu
AU - Mytnik, Leonid
TI - Strong solutions for stochastic differential equations with jumps
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 4
SP - 1055
EP - 1067
AB - General stochastic equations with jumps are studied. We provide criteria for the uniqueness and existence of strong solutions under non-Lipschitz conditions of Yamada–Watanabe type. The results are applied to stochastic equations driven by spectrally positive Lévy processes.
LA - eng
KW - stochastic equation; strong solution; pathwise uniqueness; non-Lipschitz condition
UR - http://eudml.org/doc/243441
ER -

References

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  1. [1] M. T. Barlow. One-dimensional stochastic differential equations with no strong solution. J. London Math. Soc. (2) 26 (1982) 335–347. Zbl0456.60062MR675177
  2. [2] R. F. Bass. Stochastic differential equations driven by symmetric stable processes. In Séminaire de Probabilités, XXXVI. Lecture Notes in Math. 1801 302–313. Springer, Berlin, 2003. Zbl1039.60056MR1971592
  3. [3] R. F. Bass, K. Burdzy and Z.-Q. Chen. Stochastic differential equations driven by stable processes for which pathwise uniqueness fails. Stochastic Process. Appl. 111 (2004) 1–15. Zbl1111.60038MR2049566
  4. [4] Z. F. Fu and Z. H. Li. Stochastic equations of non-negative processes with jumps. Stochastic Process. Appl. 120 (2010) 306–330. Zbl1184.60022MR2584896
  5. [5] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes, 2nd edition. North-Holland and Kodasha, Amsterdam and Tokyo, 1989. Zbl0495.60005MR1011252
  6. [6] T. Komatsu. On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations of jump type. Proc. Japan Acad. Ser. A Math. Sci. 58 (1982) 353–356. Zbl0511.60057MR683262
  7. [7] R. Situ. Theory of Stochastic Differential Equations with Jumps and Applications. Springer, Berlin, 2005. Zbl1070.60002MR2160585
  8. [8] T. Yamada and S. Watanabe. On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 (1971) 155–167. Zbl0236.60037MR278420

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