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Limit behavior of functionals of solutions of diffusion type equations


Authors: G. L. Kulinich, S. V. Kushnirenko and Yu. S. Mishura
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 92 (2015).
Journal: Theor. Probability and Math. Statist. 92 (2016), 93-107
MSC (2010): Primary 60H10; Secondary 60J60
DOI: https://doi.org/10.1090/tpms/985
Published electronically: August 10, 2016
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Abstract: The asymptotic behavior as $ T \to \infty $ of the functionals $ I (tT) $ with an appropriate normalizing factor is studied, where $ I (t) = F (\xi (t)) + \int _ {0} ^ {t} g (\xi (s)) \, dW (s) $, $ t \ge 0 $, $ F $ is a continuous function, $ g $ is a locally square integrable function, $ \xi $ is an unstable solution of the Itô stochastic differential equation $ d \xi (t) = a (\xi (t)) \, dt + dW (t) $, and $ a $ is a measurable and bounded function. We find the normalizing factor for the weak convergence of stochastic processes $ I(tT)$, $ t\ge 0$, for certain classes of these equations. The explicit form of the limit processes is established.


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Additional Information

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

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

Yu. S. Mishura
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrs’ka Street, 64/13, 01601, Kyiv, Ukraine
Email: myus@univ.kiev.ua

DOI: https://doi.org/10.1090/tpms/985
Keywords: Diffusion type processes, limit behavior of functionals, unstable solutions of stochastic differential equations
Received by editor(s): February 24, 2015
Published electronically: August 10, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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