Existence and uniqueness of solutions of stochastic differential equations with non-Lipschitz diffusion and Poisson measure

Authors:
V. P. Zubchenko and Yu. S. Mishura

Translated by:
N. N. Semenov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **80** (2009).

Journal:
Theor. Probability and Math. Statist. **80** (2010), 47-59

MSC (2000):
Primary 60H10; Secondary 60H05, 60J65

Published electronically:
August 18, 2010

MathSciNet review:
2541951

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The existence and uniqueness of a solution of a stochastic differential equation with a non-Lipschitz diffusion for cases of both centered and non-centered Poisson measures is proved. We prove that the pathwise uniqueness of a solution and the existence of a weak solution imply the existence of a strong solution for such equations.

**1.**I. I. Gihman and A. V. Skorohod,*Stokhasticheskie differentsialnye uravneniya*, Izdat. “Naukova Dumka”, Kiev, 1968 (Russian). MR**0263172**

Ĭ. Ī. Gīhman and A. V. Skorohod,*Stochastic differential equations*, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by Kenneth Wickwire; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72. MR**0346904****2.**I. I . Gikhman and A. V. Skorokhod,*Stokhasticheskie differentsialnye uravneniya i ikh prilozheniya*, “Naukova Dumka”, Kiev, 1982 (Russian). MR**678374****3.**Daniel W. Stroock and S. R. Srinivasa Varadhan,*Multidimensional diffusion processes*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 233, Springer-Verlag, Berlin-New York, 1979. MR**532498****4.**Nobuyuki Ikeda and Shinzo Watanabe,*Stochastic differential equations and diffusion processes*, 2nd ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR**1011252****5.**Toshio Yamada and Shinzo Watanabe,*On the uniqueness of solutions of stochastic differential equations.*, J. Math. Kyoto Univ.**11**(1971), 155–167. MR**0278420****6.**G. L. Kulīnīč,*The existence and uniqueness of the solution of a stochastic differential equation with martingale differential*, Teor. Verojatnost. i Primenen.**19**(1974), 169–173 (Russian, with English summary). MR**0345209****7.**L. I. Gal'chuk,*The structure of some martingales*, Proceedings of the School Seminar on the Theory of Stochastic Processes, vol. 1, Vilnius, 1975, pp. 9-33. (Russian)**8.**Vladimir Zubchenko,*Long-term returns in stochastic interest rate models*, Theory Stoch. Process.**13**(2007), no. 4, 247–261. MR**2482264**

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Additional Information

**V. P. Zubchenko**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, Kyiv National Taras Shevchenko University, Academician Glushkov Avenue, 2, Kiev 03127, Ukraine

Email:
v_zubchenko@ukr.net

**Yu. S. Mishura**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, Kyiv National Taras Shevchenko University, Academician Glushkov Avenue, 2, Kiev 03127, Ukraine

Email:
myus@univ.kiev.ua

DOI:
https://doi.org/10.1090/S0094-9000-2010-00793-9

Keywords:
Stochastic differential equation,
non-Lipschitz diffusion,
Poisson measure,
weak solution,
existence and uniqueness of solution

Received by editor(s):
February 25, 2009

Published electronically:
August 18, 2010

Additional Notes:
The authors are grateful to the European Commission for support of their investigations in the framework of the Program “Marie Curie Actions”, grant “Multifractionality” PIRSES-GA-2008-230804

Article copyright:
© Copyright 2010
American Mathematical Society