Geodesic
On strong existence and continuous dependence for solutions of one-dimensional stochastic equations with additive L\'evy noise
Teoriâ slučajnyh processov, Tome 18 (2012) no. 2, pp. 77-82.

Voir la notice de l'article provenant de la source Math-Net.Ru

One-dimensional stochastic differential equations (SDEs) with additive Lévy noise are considered. Conditions for strong existence and uniqueness of a solution are obtained. In particular, if the noise is a Lévy symmetric stable process with $\alpha\in(1;2),$ then the measurability and the boundedness of a drift term is sufficient for the existence of a strong solution. We also study the continuous dependence of the strong solution on the initial value and the drift.
Keywords: Stochastic flow, local times, differentiability with respect to initial data. 
@article{THSP_2012_18_2_a7,
     author = {A. Yu. Pilipenko},
     title = {On strong existence and continuous dependence for solutions of one-dimensional stochastic equations with additive {L\'evy} noise},
     journal = {Teori\^a slu\v{c}ajnyh processov},
     pages = {77--82},
     publisher = {mathdoc},
     volume = {18},
     number = {2},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2012_18_2_a7/}
}
TY  - JOUR
AU  - A. Yu. Pilipenko
TI  - On strong existence and continuous dependence for solutions of one-dimensional stochastic equations with additive L\'evy noise
JO  - Teoriâ slučajnyh processov
PY  - 2012
SP  - 77
EP  - 82
VL  - 18
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/THSP_2012_18_2_a7/
LA  - en
ID  - THSP_2012_18_2_a7
ER  - 
%0 Journal Article
%A A. Yu. Pilipenko
%T On strong existence and continuous dependence for solutions of one-dimensional stochastic equations with additive L\'evy noise
%J Teoriâ slučajnyh processov
%D 2012
%P 77-82
%V 18
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/THSP_2012_18_2_a7/
%G en
%F THSP_2012_18_2_a7
A. Yu. Pilipenko. On strong existence and continuous dependence for solutions of one-dimensional stochastic equations with additive L\'evy noise. Teoriâ slučajnyh processov, Tome 18 (2012) no. 2, pp. 77-82. http://geodesic.mathdoc.fr/item/THSP_2012_18_2_a7/

[1] H. Tanaka, M. Tsuchiya, S. Watanabe, “Perturbation of drift-type for Levy processes”, J. Math. Kyoto Univ., 14 (1974), 73–92 | MR | Zbl

[2] E. Priola, Pathwise uniqueness for singular SDEs driven by stable processes, 2010, arXiv: 1005.4237 | MR

[3] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer, Berlin, 1999 | MR | Zbl

[4] T. Komatsu, “On the martingale problem for generators of stable processes with perturbations”, Osaka J. of Math., 21 (1984), 113–132 | MR | Zbl

[5] N. I. Portenko, “Some perturbations of drift-type for symmetric stable processes”, Random Oper. Stoch. Equat., 2:3 (1994), 211–224 | MR | Zbl

[6] S. I. Podolynny, N. Portenko, “On multidimensional stable processes with locally unbounded drift”, Random Oper. Stoch. Equat., 3:2 (1995), 113–124 | MR | Zbl

[7] A. V. Skorokhod, Studies in the Theory of Random Processes, Addison-Wesley, Reading, Mass., 1965 | MR | Zbl

[8] A. M. Kulik, A. Yu. Pilipenko, “Nonlinear transformations of smooth measures on infinitely dimensional spaces”, Ukr. Mat. Zh., 52 (2000), 1226–1250 | DOI | MR | Zbl