Registration Number
0222U001233,
0121U111114
, R & D reports
Title
Asymptotic modes of perturbed random walks: on the border of modern and classical probability theory
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Head
Iksanov Oleksandr М., Доктор фізико-математичних наук
Registration Date
25-01-2022
Organization
Taras Shevchenko National University of Kyiv
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Object of research – iterated perturbed random walks on a general branching process tree, perpetuities, locally perturbed random walks, random walks with sticky barriers. Subject of research – asymptotics and limit theorems for the aforementioned objects. Purpose of work – the construction of a theory of perturbed random walks.
Results and novelty: The report presents the results obtained in the course of implementation of the second stage (2021) of the research project “Asymptotic regimes of perturbed random walks: on the edge of modern and classical probability”. In the present setting all the results are obtained for the first time.
In the first section a result on weak convergence of finite-dimensional distributions of the counting process in intermediate levels of iterated globally perturbed random walks is formulated.
The second section establishes various limit theorems for discounted convergent perpetuities: a law of large numbers, a functional limit theorem, and a law of the iterated logarithm. Also a functional limit theorem for the exponential functional of a random walk is obtained.
The third section introduces a skew stable L´evy process, which is a natural generalization of a skew Brownian motion, and investigates its properties. For the skew stable L´evy process functional limit theorems are proved, the resolvent is found, the connection with stochastic differential equations with local time is established, and the existence and uniqueness theorems for such equations are proved.
The fourth section proposes a construction of random walks with sticky barriers in spaces of an arbitrary dimension. The laws of large numbers and functional limit theorems are proved for hitting times of successive barriers.
Product Description
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Bohdansʹkyy Viktor Yu.
Bohun Vladyslav A.
Kondratenko Oleh О.
Kotelʹnykova Valeriya G.
Marynych Оleksandr V.
Pilipenko Andrii Yu.
Raschytov Bogdan S.
Samoylenko Ihor V.
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2022-03-09
