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Approximation of diffusion processes on solvable Lie groups by random walks. Local and quasi-local limit theorems
P. 202–206.
This note states several results on the exponential functionals of the Brownian motion and their approximations by Markov chains. Starting from M.Yor, such functionals were studied in mathematical finance. At the same time, they play a significant role in different settings: the analysis of diffusions on the class of solvable Lie groups, in particular on the group of (2 X 2) upper triangular matrices, with positive diagonal elements. The discrete random walks cannot properly describe the local structure of diffusion. However, instead of the usual local limit theorem (which is not applicable) its weaker form, namely quasi-local form is given.
Keywords: Brownian motionexponential functionalБроуновское движениеслучайное блужданиеrandom walkasian optionэкпоненциальный функционалазиатский опцион
Publication based on the results of:
Gnetov F., Konakov V., / Series arXiv "math". 2025. No. 2512.04667.
We establish a central limit theorem, a local limit theorem, and a law of large numbers for a natural
random walk on a symmetric space M of non-compact type and rank one. This class of spaces, which
includes the complex and quaternionic hyperbolic spaces and the Cayley hyperbolic plane, generalizes
the real hyperbolic space Hn. Our approach introduces ...
Added: December 5, 2025
Chaleenutthawut Y., Davydov V., Evdokimov M. et al., IEEE Access 2024 Vol. 12 P. 24843–24854
Decentralized finance (DeFi) offers a range of financial instruments and services that leverage the capabilities of web3 technology. Maker protocol, which enables users to obtain loans backed by cryptocurrencies, is one of them. Unlike traditional banks, Maker’s data is transparently recorded on the Ethereum blockchain. In this research paper, we focus on analyzing the lending ...
Added: September 4, 2024
Lyulko Y., Theory of Probability and Its Applications 2010 Vol. 54 No. 3 P. 516–525
In this paper we consider the problems of finding stochastic representations of functional F = F(ω) of a random walk. We obtain both single and multiple representations of functionals FN = max k ≦ N Sk, Fτ-a = max k ≦ τ-a Sk, where τ-a is the time of the first reaching the level-a, a ∈ N, by a random walk. ...
Added: August 28, 2024
Fernandez M. A., Mayskaya T., Nikandrova A., Games and Economic Behavior 2024 Vol. 144 P. 29–48
A rational principal delegates learning to an overconfident agent who overestimates the precision of the information he collects. The principal chooses between two contracts: commitment, in which the agent commits to the duration of learning in advance, and flexible, in which the agent decides when to stop learning in real time. When the agent is ...
Added: January 23, 2024
Safonov A. V., Agudov N. V., Krichigin A. V. et al., Journal of Statistical Mechanics: Theory and Experiment 2020 Vol. 2020 P. 024003
We propose a stochastic model for a memristive system by
generalizing known approaches and experimental results. We validate our
theoretical model by experiments carried out on a memristive device based
on Au/Ta/ZrO2(Y)/Ta2O5/TiN/Ti multilayer structure. In the framework
of the proposed model we obtain the exact analytic expressions for stationary
and nonstationary solutions. We analyze the equilibrium and non-equilibrium
steady-state distributions of ...
Added: December 14, 2022
Бородин А. Н., Journal of Mathematical Sciences 2022 Vol. 268 No. 5 P. 599–611
Added: December 6, 2022
Lyulko Y., Теория вероятностей и ее применения 2009 Т. 54 № 3 С. 580–589
В работе исследованы вопросы отыскания стохастических представлений для функционалов F=F(ω) от случайного блуждания. Получены как обычные, так и многократные представления некоторых функционалов "максимального" типа. ...
Added: May 19, 2022
Korolev A. V., Automation and Remote Control 2022 Vol. 13 No. 1 P. 483–501
Stochastic parameters are introduced into a model of network games with production and knowledge externalities. The model was formulated by V. Matveenko and A. Korolev and generalizes Romer’s two-period model. The agents’ productivities have both deterministic and Wiener components. The research represents the dynamics of a single agent and the dynamics in a triangle that ...
Added: April 22, 2022
Korolev A. V., , in: Frontiers of Dynamic Games: Game Theory and Management, St. Petersburg, 2020.: Cham: Birkhäuser, 2021. P. 167–187.
Added: April 5, 2022
Karpov I., Glazkova E., , in: Recent Trends in Analysis of Images, Social Networks and Texts. 9th International Conference, AIST 2020, Skolkovo, Moscow, Russia, October 15–16, 2020 Revised Supplementary ProceedingsVol. 12602.: Springer, 2021. P. 11–21.
The widespread of Online Social Networks and the opportunity to commercialize popular accounts have attracted a large number of automated programs, known as artificial accounts. This paper (Project repository available at http://github.com/karpovilia/botdetection) focuses on the classification of human and fake accounts on the social network, by employing several graph neural networks, to efficiently encode attributes and ...
Added: June 19, 2021
Korolev A. V., Математическая теория игр и ее приложения 2021 № 1 С. 102–129
In this paper, stochastic parameters are introduced into the network games model with production and knowledges externalities. This model was formulated by V. Matveenko and A. Korolev and generalized two-period Romer model. Agents' productivities have deterministic and Wiener components. The research represents the dynamics of a single agent and the dynamics in a triangle which ...
Added: May 15, 2021
Korolev A. V., , in: Frontiers of Dynamic Games Game Theory and Management, St. Petersburg, 2019.: Birkhauser/Springer, 2020. Ch. 6 P. 65–85.
In this paper we introduce stochastic parameters into the network game
model with production and knowledge externalities. This model was proposed
by V. Matveenko and A. Korolev as a generalization of the two-period Romer
model. Agents differ in their productivities which have deterministic and stochastic
(Wiener) components. We study the dynamics of a single agent and the dynamics
of a ...
Added: November 30, 2020
Birkhauser/Springer, 2020.
The content of this volume is mainly based on selected talks that were given at the
“International Meeting on Game Theory (ISDG12-GTM2019),” as joint meeting of
“12th International ISDG Workshop” and “13th International Conference on Game
Theory and Management,” held in St. Petersburg, Russia on July 03–05, 2019. The
meeting was organized by St. Petersburg State University and International ...
Added: November 30, 2020
Nesterenko A., Khametov V., Труды Карельского научного центра Российской академии наук 2021 № 6 С. 49–58
В статье приведено решение задач об оптимальной остановке в ситуации, когда наблюдается случайное блуждание, а функция полезности наблюдателя --- экспоненциальная (конечный и бесконечный горизонт). Для этих задач найдено:
i) явное решение уравнения Беллмана;
ii) граница, разделяющая внутренность области остановки от области продолжения наблюдений;
iii) оптимальное правило остановки. ...
Added: November 27, 2020
A. A, Gushchin, Urusov M. A., Russian Mathematical Surveys 2019 Vol. 74 No. 5 P. 953–955
We prove that a right-continuous integrable stochastic process admits a minimal embedding in the standard Brownian motion if and only if it is a submartingale or supermartingale. ...
Added: July 20, 2020
Konakov V., Menozzi S., Molchanov S., , in: Contemporary MathematicsVol. 739: Probabilistic Methods in Geometry, Topology and Spectral Theory.: AMS, 2019. P. 97–124.
This paper is concerned with Random walk approximations of the
Brownian motion on the Affine group Aff(R). We are in particular interested
in the case where the innovations are discrete. In this framework, the return
probabilities of the walk have fractional exponential decay in large time, as
opposed to the polynomial one of the continuous object. We prove that ...
Added: December 30, 2019
Molchanov S., Vainberg B., SIAM Journal on Mathematical Analysis 2019 Vol. 51 No. 3 P. 1824–1835
Symmetric random walks in $R^d$ and $Z^d$ are considered. It is assumed that the jump distribution density has moderate tails, i.e., several density moments are finite, including the second one. The global (for all $x$ and $t$) asymptotic behavior at infinity of the transition probability (fundamental solution of the corresponding parabolic convolution operator) is found. ...
Added: November 14, 2019
А. А. Гущин, Урусов М. А., Успехи математических наук 2019 Т. 74 № 5 С. 185–186
Доказано, что непрерывный справа интегрируемый случайный процесс допускает минимальное вложение в стандартное броуновское движение тогда и только тогла, когда он является субмартингалом или супермартингалом. ...
Added: September 28, 2019
V.L. Kreps, Automation and Remote Control 2019 Vol. 80 No. 2 P. 362–379
Using a simplified multistage bidding model with asymmetrically informed agents, De Meyer and Saley [17] demonstrated an idea of endogenous origin of the Brownian component in the evolution of prices on stock markets: random price fluctuations may be caused by strategic randomization of “insiders.” The model is reduced to a repeated game with incomplete information. ...
Added: May 7, 2019