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Asymptotic behavior of the martingale type integral functionals for unstable solutions to stochastic differential equations


Authors: G. L. Kulinich, S. V. Kushnirenko and Yu. S. Mishura
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 90 (2014).
Journal: Theor. Probability and Math. Statist. 90 (2015), 115-126
MSC (2010): Primary 60H10; Secondary 60F17
DOI: https://doi.org/10.1090/tpms/953
Published electronically: August 7, 2015
MathSciNet review: 3242024
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider functionals of the type $ \int _ {0} ^ {t} g (\xi (s)) \, dW (s) $, $ t \ge 0 $. Here $ g $ is a real valued and locally square integrable function, $ \xi $ is a unique strong solution of the Itô stochastic differential equation $ d \xi (t) = a (\xi (t)) \, dt + dW (t) $, $ a $ is a measurable real valued bounded function such that $ \vert xa (x) \vert \le C $. The behavior of these functionals is studied as $ t \to \infty $. The appropriate normalizing factor and the explicit form of the limit random variable are established.


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Additional Information

G. L. Kulinich
Affiliation: Department of General Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine
Email: zag_mat@univ.kiev.ua

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

Yu. S. Mishura
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Volodymyrs’ka Street, 64, Kyiv 01601, Ukraine
Email: myus@univ.kiev.ua

DOI: https://doi.org/10.1090/tpms/953
Keywords: It\^o stochastic differential equations, unstable solutions, asymptotic behavior of martingale type functionals
Received by editor(s): March 14, 2014
Published electronically: August 7, 2015
Article copyright: © Copyright 2015 American Mathematical Society