arXiv subjects
Publications and source records attributed to Alexey Kuznetsov.
We give a direct evaluation of a curious integral identity, which follows from the work of Ismail and Valent on the Nevanlinna parametrization of solutions to a certain indeterminate moment problem.
Exponential functionals of Brownian motion have been extensively studied in financial and insurance mathematics due to their broad applications, for example, in the pricing of Asian options. The Black-Scholes model is appealing because of mathematical tractability, yet empirical evidence shows that geometric Brownian motion does not adequately capture features of market equity returns. One popular alternative for modeling equity returns consists in replacing the geometric Brownian motion by an exponential of a Levy process. In this paper we use this latter model to study variable annuity guaranteed benefits and to compute explicitly the distribution of certain exponential functionals.
We describe a highly efficient numerical scheme for finding two-sided bounds for the eigenvalues of the fractional Laplace operator (-Delta)^{alpha/2} in the unit ball D in R^d, with a Dirichlet condition in the complement of D. The standard Rayleigh-Ritz variational method is used for the upper bounds, while the lower bounds involve the less-known Aronszajn method of intermediate problems. Both require explicit expressions for the fractional Laplace operator applied to a linearly dense set of functions in L^2(D). We use appropriate Jacobi-type orthogonal polynomials, which were studied in a companion paper "Fractional Laplace operator and Meijer G-function". Our numerical scheme can be applied analytically when polynomials of degree two are involved. This is used to partially resolve the conjecture of T. Kulczycki, which claims that the second smallest eigenvalue corresponds to an antisymmetric function: we prove that this is the case when either d <= 2 and 0 < alpha <= 2, or d <= 9 and alpha = 1, and we provide strong numerical evidence for d <= 9 and general alpha in (0,2].
The Nevanlinna parametrization establishes a bijection between the class of all measures having a prescribed set of moments and the class of Pick functions. The fact that all measures constructed through the Nevanlinna parametrization have identical moments follows from the theory of orthogonal polynomials and continued fractions. In this paper we explore the opposite direction: we construct a set of measures and we show that they all have identical moments, and then we establish a Nevanlinna-type parametrization for this set of measures. Our construction does not require the theory of orthogonal polynomials and it exposes the analytic structure behind the Nevanlinna parametrization.
Tail dependence refers to clustering of extreme events. In the context of financial risk management, the clustering of high-severity risks has a devastating effect on the well-being of firms and is thus of pivotal importance in risk analysis.When it comes to quantifying the extent of tail dependence, it is generally agreed that measures of tail dependence must be independent of the marginal distributions of the risks but rather solely copula-dependent. Indeed, all classical measures of tail dependence are such, but they investigate the amount of tail dependence along the main diagonal of copulas, which has often little in common with the concentration of extremes in the copulas' domain of definition.In this paper we urge that the classical measures of tail dependence may underestimate the level of tail dependence in copulas. For the Gaussian copula, however, we prove that the classical measures are maximal. The implication of the result is two-fold: On the one hand, it means that in the Gaussian case, the (weak) measures of tail dependence that have been reported and used are of utmost prudence, which must be a reassuring news for practitioners. On the other hand, it further encourages substitution of the Gaussian copula with other copulas that are more tail dependent.
Identities involving finite sums of products of hypergeometric functions and their duals have been studied since 1930s. Recently Beukers and Jouhet have used an algebraic approach to derive a very general family of duality relations. In this paper we provide an alternative way of obtaining such results. Our method is very simple and it is based on the non-local derangement identity.
In this paper, we produce synthetic radio views on simulated flux ropes in the solar corona, where finite-beta magnetohydrodynamic (MHD) simulations serve to mimic the flux rope formation stages, as well as their stable endstates. These endstates represent twisted flux ropes where balancing Lorentz forces, gravity and pressure gradients determine the full thermodynamic variation throughout the flux rope. The obtained models are needed to quantify radiative transfer in radio bands, and allow us to contrast weak to strong magnetic field conditions. Field strengths of up to 100 G in the flux rope yield the radio views dominated by optically thin free-free emission. The forming flux rope shows clear morphological changes in its emission structure as it deforms from an arcade to a flux rope, both on disk and at the limb. For an active region filament channel with a field strength of up to 680 G in the flux rope, gyroresonance emission (from the third-fourth gyrolayers) can be detected and even dominates over free-free emission at the frequencies of up to 7 GHz. Finally, we also show synthetic views on a simulated filament embedded within a (weak-field) flux rope, resulting from an energetically consistent MHD simulation. For this filament, synthetic views at the limb show clear similarities with actual observations, and the transition from optically thick (below 10 GHz) to optically thin emission can be reproduced. On the disk, its dimension and temperature conditions are as yet not realistic enough to yield the observed radio brightness depressions.
Lévy processes with completely monotone jumps appear frequently in various applications of probability. For example, all popular stock price models based on Lévy processes (such as the Variance Gamma, CGMY/KoBoL and Normal Inverse Gaussian) belong to this class. In this paper we continue the work started in [Int. J. Theor. Appl. Finance 13 (2010) 63-91, Quant. Finance 10 (2010) 629-644] and develop a simple yet very efficient method for approximating processes with completely monotone jumps by processes with hyperexponential jumps, the latter being the most convenient class for performing numerical computations. Our approach is based on connecting Lévy processes with completely monotone jumps with several areas of classical analysis, including Padé approximations, Gaussian quadrature and orthogonal polynomials.
We significantly expand the number of functions whose image under the fractional Laplace operator can be computed explicitly. In particular, we show that the fractional Laplace operator maps Meijer G-functions of |x|^2, or generalized hypergeometric functions of -|x|^2, multiplied by a solid harmonic polynomial, into the same class of functions. As one important application of this result, we produce a complete system of eigenfunctions of the operator (1-|x|^2)_+^{alpha/2} (-Delta)^{alpha/2} with the Dirichlet boundary conditions outside of the unit ball. The latter result will be used to estimate the eigenvalues of the fractional Laplace operator in the unit ball in a companion paper "Eigenvalues of the fractional Laplace operator in the unit ball".
Radio observations of solar flares often reveal various periodic or quasi-periodic oscillations. Most likely, these oscillations are caused by magnetohydrodynamic (MHD) oscillations of flaring loops which modulate the emission. Interpretation of the observations requires comparing them with simulations. We simulate the gyrosynchrotron radio emission from a semi-circular (toroidal-shaped) magnetic loop containing sausage-mode MHD oscillations. The aim is to detect the observable signatures specific to the considered MHD mode and to study their dependence on the various source parameters. The MHD waves are simulated using a linear three-dimensional model of a magnetized plasma cylinder; both standing and propagating waves are considered. The curved loop is formed by replicating the MHD solutions along the plasma cylinder and bending the cylinder; this model allows us to study the effect of varying the viewing angle along the loop. The radio emission is simulated using a three-dimensional model and its spatial and temporal variations are analyzed. We consider several loop orientations and different parameters of the magnetic field, plasma, and energetic electrons in the loop. In the model with low plasma density, the intensity oscillations at all frequencies are synchronous (with the exception of a narrow spectral region below the spectral peak). In the model with high plasma density, the emission at low frequencies (where the Razin effect is important) oscillates in anti-phase with the emissions at higher frequencies. The oscillations at high and low frequencies are more pronounced in different parts of the loop (depending on the loop orientation). The layers where the line-of-sight component of the magnetic field changes sign can produce additional peculiarities in the oscillation patterns.
The search for a canonical set of eigenvectors of the discrete Fourier transform has been ongoing for more than three decades. The goal is to find an orthogonal basis of eigenvectors which would approximate Hermite functions -- the eigenfunctions of the continuous Fourier transform. This eigenbasis should also have some degree of analytical tractability and should allow for efficient numerical computations. In this paper we provide a partial solution to these problems. First, we construct an explicit basis of (non-orthogonal) eigenvectors of the discrete Fourier transform, thus extending the results of [7]. Applying the Gramm-Schmidt orthogonalization procedure we obtain an orthogonal eigenbasis of the discrete Fourier transform. We prove that the first eight eigenvectors converge to the corresponding Hermite functions, and we conjecture that this convergence result remains true for all eigenvectors.
There exist only a few known examples of subordinators for which the transition probability density can be computed explicitly along side an expression for its Lévy measure and Laplace exponent. Such examples are useful in several areas of applied probability, for example, they are used in mathematical finance for modeling stochastic time change, they appear in combinatorial probability to construct sampling formulae, which in turn is related to a variety of issues in the theory of coalescence models, moreover, they have also been extensively used in the potential analysis of subordinated Brownian motion in dimension greater than or equal to 2. In this paper, we show that Kendall's classic identity for spectrally negative Lévy processes can be used to construct new families of subordinators with explicit transition probability semigroups. We describe the properties of these new subordinators and emphasise some interesting connections with explicit and previously unknown Laplace transform identities and with complete monotonicity properties of certain special functions.
We introduce a class of reinforcement models where, at each time step $t$, one first chooses a random subset $A_t$ of colours (independent of the past) from $n$ colours of balls, and then chooses a colour $i$ from this subset with probability proportional to the number of balls of colour $i$ in the urn raised to the power $α>1$. We consider stability of equilibria for such models and establish the existence of phase transitions in a number of examples, including when the colours are the edges of a graph, a context which is a toy model for the formation and reinforcement of neural connections.
Hiary [3] has presented an algorithm which allows to evaluate the truncated theta function $\sum_{k=0}^n \exp(2πı(zk+τk^2))$ to within $\pm ε$ in $O(\ln(\tfrac{n}ε)^κ)$ arithmetic operations for any real $z$ and $τ$. This remarkable result has many applications in Number Theory, in particular it is the crucial element in Hiary's algorithm for computing $ζ(\tfrac{1}{2}+ıt)$ to within $\pm t^{-λ}$ in $O_λ(t^{\frac{1}{3}}\ln(t)^κ)$ arithmetic operations, see [2]. We present a significant simplification of Hiary's algorithm for evaluating the truncated theta function. Our method avoids the use of the Poisson summation formula, and substitutes it with an explicit identity involving the Mordell integral. This results in an algorithm which is efficient, conceptually simple and easy to implement.
We investigate in detail the 21 May 2004 flare using simultaneous observations of the Nobeyama Radioheliograph, Nobeyama Radiopolarimeters, Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) and Solar and Heliospheric Observatory (SOHO). The flare images in different spectral ranges reveal the presence of a well-defined single flaring loop in this event. We have simulated the gyrosynchrotron microwave emission using the recently developed interactive IDL tool GX Simulator. By comparing the simulation results with the observations, we have deduced the spatial and spectral properties of the non-thermal electron distribution. The microwave emission has been found to be produced by the high-energy electrons ($>100$ keV) with a relatively hard spectrum ($δ\simeq 2$); the electrons were strongly concentrated near the loop top. At the same time, the number of high-energy electrons near the footpoints was too low to be detected in the RHESSI images and spatially unresolved data. The SOHO Extreme-ultraviolet Imaging Telescope images and the low-frequency microwave spectra suggest the presence of an extended "envelope" of the loop with lower magnetic field. Most likely, the energetic electron distribution in the considered flare reflects the localized (near the loop top) particle acceleration (injection) process accompanied by trapping and scattering.
For any two-sided jumping $α$-stable process, where $1 < α< 2$, we find an explicit identity for the law of the first hitting time of the origin. This complements existing work in the symmetric case and the spectrally one-sided case; cf. Yano-Yano-Yor (2009) and Cordero (2010), and Peskir (2008) respectively. We appeal to the Lamperti-Kiu representation of Chaumont-Pantí-Rivero (2011) for real-valued self-similar Markov processes. Our main result follows by considering a vector-valued functional equation for the Mellin transform of the integrated exponential Markov additive process in the Lamperti-Kiu representation. We conclude our presentation with some applications.
The Gaver-Stehfest algorithm for numerical inversion of Laplace transform was developed in the late 1960s. Due to its simplicity and good performance it is becoming increasingly more popular in such diverse areas as Geophysics, Operations Research and Economics, Financial and Actuarial Mathematics, Computational Physics and Chemistry. Despite the large number of applications and numerical studies, this method has never been rigorously investigated. In particular, it is not known whether the Gaver-Stehfest approximations converge and what is the rate of convergence. In this paper we answer the first of these two questions: We prove that the Gaver-Stehfest approximations converge for functions of bounded variation and functions satisfying an analogue of Dini criterion.
An absolutely convergent double series representation for the density of the supremum of $α$-stable Levy process is given in [3, Theorem 2] for almost all irrational $α$. This result cannot be made stronger in the following sense: the series does not converge absolutely when $α$ belongs to a certain subset of irrational numbers of Lebesgue measure zero (see [6, Theorem 2]). Our main result in this note shows that for every irrational $α$ there is a way to rearrange the terms of the double series, so that it converges to the density of the supremum. We show how one can establish this stronger result by introducing a simple yet non-trivial modification in the original proof of [3,Theorem 2].