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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
DOI: https://doi.org/10.1090/S0094-9000-2010-00793-9
Published electronically: August 18, 2010
MathSciNet review: 2541951
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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.


References [Enhancements On Off] (What's this?)

  • 1. I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, Naukova Dumka, Kiev, 1968; English transl., Springer-Verlag, New York-Heidelberg, 1972.MR 0263172 (41:7777); MR 0346904 (49:11625)
  • 2. I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations and their Applications, Naukova Dumka, Kiev, 1982. (Russian) MR 678374 (84j:60003)
  • 3. D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, Berlin, 1979. MR 532498 (81f:60108)
  • 4. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, second edition, North-Holland, Amsterdam, 1989. MR 1011252 (90m:60069)
  • 5. T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971), no. 1, 155-167. MR 0278420 (43:4150)
  • 6. G. L. Kulinich, On existence and uniqueness of a solution of a stochastic differential equation with martingale differential, Teor. Veroyatnost. Primenen. XIX (1974), no. 1, 169-173; English transl. in Theory Probab. Appl. 19 (1974-1975), no. 1, 168-171. MR 0345209 (49:9948)
  • 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. V. Zubchenko, Long-term returns in stochastic interest rate models, Theory Stoch. Process. 13 (2007), no. 4, 247-261. MR 2482264 (2001h:60188)

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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

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