arXiv subjects

Andrey Sarantsev

Publications and source records attributed to Andrey Sarantsev.

51 records · Page 3Linked to original sources

arXiv2017

Stationary Gap Distributions for Infinite Systems of Competing Brownian Particles

Consider the infinite Atlas model: a semi-infinite collection of particles driven by independent standard Brownian motions with zero drifts, except for the bottom-ranked particle which receives unit drift. We derive a continuum one-parameter family of product-of-exponentials stationary gap distributions, with exponentially growing density at infinity. This result shows that there are infinitely many stationary gap distributions for the Atlas model, and hence resolves a conjecture of Pal and Pitman (2008) in the negative. This result is further generalized for infinite systems of competing Brownian particles with generic rank-based drifts.

math.PR↗

arXiv2017

Yet Another Condition for Absence of Collisions for Competing Brownian Particles

Consider a finite system of rank-based competing Brownian particles, where the drift and diffusion of each particle depend only on its current rank relative to other particles. We present a simple sufficient condition for absence of multiple collisions of a given order, continuing the earlier work by Bruggeman and Sarantsev (2015). Unlike in that paper, this new condition works even for infinite systems.

math.PR↗

arXiv2016

Explicit Rates of Exponential Convergence for Reflected Jump-Diffusions on the Half-Line

Consider a reflected jump-diffusion on the positive half-line. Assume it is stochastically ordered. We apply the theory of Lyapunov functions and find explicit estimates for the rate of exponential convergence to the stationary distribution, as time goes to infinity. This continues the work of Lund, Meyn and Tweedie (1996). We apply these results to systems of two competing Levy particles with rank-dependent dynamics.

math.PR↗

arXiv2016

Penalty Method for Reflected Diffusions on the Half-Line

Consider a reflected diffusion on the positive half-line. We approximate it by solutions of stochastic differential equations using the penalty method: We emulate the "hard barrier" of reflection by a "soft barrier" of a large drift coefficient, which compells the diffusion to return to the positive half-line. The main tool of the proof is convergence of scale functions.

math.PR↗

arXiv2016

Infinite Systems of Competing Brownian Particles

Consider a system of infinitely many Brownian particles on the real line. At any moment, these particles can be ranked from the bottom upward. Each particle moves as a Brownian motion with drift and diffusion coefficients depending on its current rank. The gaps between consecutive particles form the (infinite-dimensional) gap process. We find a stationary distribution for the gap process. We also show that if the initial value of the gap process is stochastically larger than this stationary distribution, this process converges back to this distribution as time goes to infinity. This continues the work by Pal and Pitman (2008). Also, this includes infinite systems with asymmetric collisions, similar to the finite ones from Karatzas, Pal and Shkolnikov (2016).

math.PR↗

arXiv2016

Diverse market models of competing Brownian particles with splits and mergers

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.

math.PR↗

arXiv2016

Comparison Techniques for Competing Brownian Particles

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.

math.PR↗

arXiv2016

Multiple Collisions in Systems of Competing Brownian Particles

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. We find a sufficient condition for a.s. absence of a total collision (when all particles collide) and of other types of collisions, say of the three lowest-ranked particles. This continues the work of Ichiba, Karatzas, Shkolnikov (2013) and Sarantsev (2015).

math.PR↗

arXiv2016

Reflected Brownian Motion in a Convex Polyhedral Cone: Tail Estimates for the Stationary Distribution

Consider an 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 the exponential rate, as time goes to infinity. 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.

math.PR↗

arXiv2015

On the nature of $a_1(1420)$

The resonance-like signal with axial-vector quantum numbers $J^{PC}=1^{++}$ at a mass of $1420\,$MeV and a width of $140\,$MeV, recently observed by the COMPASS and VES experiments in the $f_0(980)π$ final state and tentatively called $a_1(1420)$, is discussed. Instead of a genuine new meson, we interpret this signal as a dynamical effect due to a singularity (branching point) in the triangle diagram formed by the processes $a_1(1260) \to K^\star \bar{K}$, $K^\star\to Kπ$, and $K \bar{K} \to f_0(980)$ (+ c.c). The amplitude for this diagram is calculated. The result exhibits a peak in the intensity with a sharp phase motion with respect to the dominant $a_1(1260) \to ρπ$ $S$-wave decay, in good agreement with the data. The branching ratio of $a_1(1260)\to f_0(980)π$ via the triangle diagram is estimated and compared to the dominant decay $a_1(1260)\toρπ$.

hep-ph↗

arXiv2015

Triple and Simultaneous Collisions of Competing Brownian Particles

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).

math.PR↗

arXiv2013

On a Class of Diverse Market Models

A market model in Stochastic Portfolio Theory is a finite system of strictly positive stochastic processes. Each process represents the capitalization of a certain stock. If at any time no stock dominates almost the entire market, which means that its share of total market capitalization is not very close to one, then the market is called diverse. There are several ways to outperform diverse markets and get an arbitrage opportunity, and this makes these markets interesting. A feature of real-world markets is that stocks with smaller capitalizations have larger drift coefficients. Some models, like the Volatility-Stabilized Model, try to capture this property, but they are not diverse. In an attempt to combine this feature with diversity, we construct a new class of market models. We find simple, easy-to-test sufficient conditions for them to be diverse and other sufficient conditions for them not to be diverse.

math.PR↗

arXiv2013

Tail Asymptotic of Sum and Product of Random Variables with Applications in the Theory of Extremes of Conditionally Gaussian Processes

We consider two independent random variables with the given tail asymptotic (e.g. power or exponential). We find tail asymptotic for their sum and product. This is done by some cumbersome but purely technical computations and requires the use of the Laplace method for asymptotic of integrals. We also recall the results for asymptotic of a self-similar locally stationary centered Gaussian process plus a deterministic drift; and we find the asymptotic for the same probability after multiplying the drift by a random variable, which is independent of this process. Keywords: tail asymptotic, Laplace method, self-similar processes, Gaussian processes, locally stationary processes.

math.PR↗

arXiv2013

Statistical Investigation of Increments of Currency Rates Logarithms

We consider the currency rates dynamics for 12 currencies, including dollar and euro, with respect to Russian rouble. We prove that the Samuelson model (geometric Brownian motion) is not suitable for this dynamics. We also prove that another model (with inverse Gaussian increments of logarithms) is not appropriate for this situation. We point out the difference in behavior of different currencies, and the difference in behavior before and during the financial crisis which began in 2008.

math.ST↗

arXiv2013

On a Generalization of Bernoulli and Euler Numbers

We introduce a series of numbers which serve as a generalization of Bernoulli, Euler numbers and binomial coefficients. Their properties are applied to solve a probability problem and suggest a statistical test for independence and identical distribution of random variables.

math.CO↗