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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

 

Properties of solutions of stochastic differential equations with nonhomogeneous coefficients and non-Lipschitz diffusion


Authors: Yu. S. Mishura, S. V. Posashkova and G. M. Shevchenko
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 79 (2008).
Journal: Theor. Probability and Math. Statist. 79 (2009), 117-126
MSC (2000): Primary 60H10; Secondary 91B28
Published electronically: December 28, 2009
MathSciNet review: 2494541
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Abstract: Properties of solutions of stochastic differential equations with nonhomogeneous coefficients and non-Lipschitz diffusion are studied in the paper. Conditions on the coefficients of an equation are obtained ensuring that a solution does not vanish over a finite time interval in the case of the diffusion $ \sigma(t)\sqrt{x}$. We prove a limit theorem that solutions continuously depend on the parameter $ n$ in the space $ L_1(\mathsf{P})$ for a sequence of stochastic differential equations with nonhomogeneous coefficients and non-Lipschitz diffusion.


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Additional Information

Yu. S. Mishura
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email: myus@univ.kiev.ua

S. V. Posashkova
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email: revan1988@gmail.com

G. M. Shevchenko
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email: zhora@univ.kiev.ua

DOI: http://dx.doi.org/10.1090/S0094-9000-09-00774-1
PII: S 0094-9000(09)00774-1
Keywords: Stochastic differential equations, non-Lipschitz diffusion, Cox--Ingersoll--Ross model, continuous dependence on a parameter
Received by editor(s): March 12, 2008
Published electronically: December 28, 2009
Article copyright: © Copyright 2009 American Mathematical Society