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Asymptotic behavior of integral functionals of unstable solutions of one-dimensional Itô stochastic differential equations


Authors: G. L. Kulinich, S. V. Kushnirenko and Y. S. Mishura
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 89 (2013).
Journal: Theor. Probability and Math. Statist. 89 (2014), 101-114
MSC (2010): Primary 60H10
DOI: https://doi.org/10.1090/S0094-9000-2015-00938-8
Published electronically: January 26, 2015
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider one-dimensional stochastic differential equations with a homogeneous drift and unit diffusion. The drift is such that a unique strong solution is unstable. An explicit form of the normalizing factor is established for certain integral functionals of the unstable solution for which the weak convergence to a limiting process holds. As a result, we get a new class of limiting processes that are the functionals of Bessel diffusion processes.


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Additional Information

G. L. Kulinich
Affiliation: Department of General Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, Kyiv, Ukraine, 01601
Email: zag$_$mat@univ.kiev.ua

S. V. Kushnirenko
Affiliation: Department of General Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, Kyiv, Ukraine, 01601
Email: bksv@univ.kiev.ua

Y. S. Mishura
Affiliation: Department of Probability Theory, Statistics and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 64/13 Volodymyrska Street, Kyiv, Ukraine, 01601
Email: yumishura@gmail.com

DOI: https://doi.org/10.1090/S0094-9000-2015-00938-8
Keywords: It\^o stochastic differential equation, unstable solution, asymptotic behavior of integral functionals
Received by editor(s): December 12, 2012
Published electronically: January 26, 2015
Article copyright: © Copyright 2015 American Mathematical Society