Author
Andrey Sarantsev
Other affiliations: Carnegie Mellon University, University of California, Santa Barbara, University of Washington
Bio: Andrey Sarantsev is an academic researcher from University of Nevada, Reno. The author has contributed to research in topics: Brownian motion & Stationary distribution. The author has an hindex of 11, co-authored 75 publications receiving 440 citations. Previous affiliations of Andrey Sarantsev include Carnegie Mellon University & University of California, Santa Barbara.
Papers
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TL;DR: Ichiba, Karatzas, Shkolnikov as mentioned in this paper showed a necessary and sufficient condition for a.s. absense of triple and simultaneous collisions in the case of asymmetric collisions, when the local time of collision between the particles is split unevenly between them.
Abstract: Consider a finite system of competing Brownian particles on the real line. Each particle moves as a Brownian motion, with drift and diffusion coefficients depending only on its current rank relative to the other particles. A triple collision occurs if three particles are at the same position at the same moment. A simultaneous collision occurs if at a certain moment, there are two distinct pairs of particles such that in each pair, both particles occupy the same position. These two pairs of particles can overlap, so a triple collision is a particular case of a simultaneous collision. We find a necessary and sufficient condition for a.s. absense of triple and simultaneous collisions, continuing the work of Ichiba, Karatzas, Shkolnikov (2013). Our results are also valid for the case of asymmetric collisions, when the local time of collision between the particles is split unevenly between them; these systems were introduced in Karatzas, Pal, Shkolnikov
48 citations
TL;DR: In this paper, a multidimensional obliquely reflected Brownian motion in a convex polyhedral cone is considered and sufficient conditions for existence of a stationary distribution and convergence to this distribution at an exponential rate, as time goes to infinity.
Abstract: Consider a multidimensional obliquely reflected Brownian motion in the positive orthant, or, more generally, in a convex polyhedral cone. We find sufficient conditions for existence of a stationary distribution and convergence to this distribution at an exponential rate, as time goes to infinity, complementing the results of Dupuis and Williams (Ann Probab 22(2):680–702, 1994) and Atar et al. (Ann Probab 29(2):979–1000, 2001). We also prove that certain exponential moments for this distribution are finite, thus providing a tail estimate for this distribution. Finally, we apply these results to systems of rank-based competing Brownian particles, introduced in Banner et al. (Ann Appl Probab 15(4):2296–2330, 2005).
39 citations
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TL;DR: In this article, a necessary and sufficient condition for a.s. absense of triple and simultaneous collisions was established for the case of asymmetric collisions, when the local time of collision between the particles is split unevenly between them.
Abstract: Consider a finite system of competing Brownian particles on the real line. Each particle moves as a Brownian motion, with drift and diffusion coefficients depending only on its current rank relative to the other particles. A triple collision occurs if three particles are at the same position at the same moment. A simultaneous collision occurs if at a certain moment, there are two distinct pairs of particles such that in each pair, both particles occupy the same position. These two pairs of particles can overlap, so a triple collision is a particular case of a simultaneous collision. We find a necessary and sufficient condition for a.s. absense of triple and simultaneous collisions, continuing the work of Ichiba, Karatzas, Shkolnikov (2013). Our results are also valid for the case of asymmetric collisions, when the local time of collision between the particles is split unevenly between them; these systems were introduced in Karatzas, Pal, Shkolnikov (2012).
36 citations
TL;DR: In this article, the authors studied a regulatory break-up model with a fluctuating number of companies, where each company has a capitalization whose logarithm behaves as a Brownian motion with drift and diffusion coefficients depending on its current rank.
Abstract: We study models of regulatory breakup, in the spirit of Strong and Fouque [Ann. Finance 7 (2011) 349-374] but with a fluctuating number of companies. An important class of market models is based on systems of competing Brownian particles: each company has a capitalization whose logarithm behaves as a Brownian motion with drift and diffusion coefficients depending on its current rank. We study such models with a fluctuating number of companies: If at some moment the share of the total market capitalization of a company reaches a fixed level, then the company is split into two parts of random size. Companies are also allowed to merge, when an exponential clock rings. We find conditions under which this system is nonexplosive (i.e., the number of companies remains finite at all times) and diverse, yet does not admit arbitrage opportunities.
28 citations
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TL;DR: In this article, the authors consider a finite system of Brownian particles on the real line and show that if a few particles from the top of the line are removed, the gaps between adjacent particles become stochastically larger, the local times of collision between neighboring particles become smaller, and the remaining particles shift upward.
Abstract: Consider a finite system of Brownian particles on the real line. Each particle has drift and diffusion coefficients depending on its current rank relative to other particles, as in Karatzas, Pal and Shkolnikov (2012). We prove some comparison results for these systems. As an example, we show that if we remove a few particles from the top, then the gaps between adjacent particles become stochastically larger, the local times of collision between adjacent particles become stochastically smaller, and the remaining particles shift upward, in the sense of stochastic ordering.
26 citations
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Journal Article•
28,685 citations
TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.
5,689 citations
01 May 1975
TL;DR: The Fundamentals of Queueing Theory, Fourth Edition as discussed by the authors provides a comprehensive overview of simple and more advanced queuing models, with a self-contained presentation of key concepts and formulae.
Abstract: Praise for the Third Edition: "This is one of the best books available. Its excellent organizational structure allows quick reference to specific models and its clear presentation . . . solidifies the understanding of the concepts being presented."IIE Transactions on Operations EngineeringThoroughly revised and expanded to reflect the latest developments in the field, Fundamentals of Queueing Theory, Fourth Edition continues to present the basic statistical principles that are necessary to analyze the probabilistic nature of queues. Rather than presenting a narrow focus on the subject, this update illustrates the wide-reaching, fundamental concepts in queueing theory and its applications to diverse areas such as computer science, engineering, business, and operations research.This update takes a numerical approach to understanding and making probable estimations relating to queues, with a comprehensive outline of simple and more advanced queueing models. Newly featured topics of the Fourth Edition include:Retrial queuesApproximations for queueing networksNumerical inversion of transformsDetermining the appropriate number of servers to balance quality and cost of serviceEach chapter provides a self-contained presentation of key concepts and formulae, allowing readers to work with each section independently, while a summary table at the end of the book outlines the types of queues that have been discussed and their results. In addition, two new appendices have been added, discussing transforms and generating functions as well as the fundamentals of differential and difference equations. New examples are now included along with problems that incorporate QtsPlus software, which is freely available via the book's related Web site.With its accessible style and wealth of real-world examples, Fundamentals of Queueing Theory, Fourth Edition is an ideal book for courses on queueing theory at the upper-undergraduate and graduate levels. It is also a valuable resource for researchers and practitioners who analyze congestion in the fields of telecommunications, transportation, aviation, and management science.
2,562 citations
TL;DR: In this paper, applied probability and queuing in the field of applied probabilistic analysis is discussed. But the authors focus on the application of queueing in the context of road traffic.
Abstract: (1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.
1,121 citations