Estimates eigenvalues using Bessel functions on manifolds.
problem Estimating eigenvalues of differential operators on manifolds.
method Using mean value lemma and curvature assumptions, derive differential inequalities involving Bessel functions.
result Establishes new estimates for eigenvalues involving positive roots of Bessel functions.
Study Hardy identities and inequalities on Cartan-Hadamard manifolds.
problem Existence and nonexistence of extremal functions in Hardy inequalities.
method Using the notion of a Bessel pair, we derive Hardy identities and inequalities.
result Established several Hardy type inequalities with improvements and understandings.
Orthogonal random features approximate a Bessel kernel, offering sharper bounds than random Fourier features.
problem Approximating Gaussian kernel efficiently for large datasets.
method Use of Haar orthogonal matrices to construct orthogonal random features and analyze their bias and variance.
result Orthogonal random features approximate a Bessel kernel, not the Gaussian kernel, with sharper bounds.
Derivatives of sub-Riemannian geodesics are always Lp-Hölder continuous.
problem Smoothness of sub-Riemannian geodesics
method Proving Lp-Hölder continuity of derivatives result Derivatives of sub-Riemannian geodesics are Lp-Hölder continuous In this paper we study the problem of deriving further Sobolev inequalities from a given Sobolev inequality. We use several different methods, including Bessel potentials and Riesz transforms. We apply the results to the Ricci flow to extend the author's results on the W1,2 Sobolev inequality along the Ricci flow …
Paper proves smoothness for variational inference, giving convergence guarantees.
problem Proving convergence guarantees for black-box variational inference.
method Describes gradients in an inner-product space, using Bessel's inequality.
result Objective is M-Lipschitz smooth if target is, excluding entropy.
Constructs surfaces with constant mean curvature from Bessel equation.
problem Creating surfaces with constant mean curvature.
method From the Bessel equation, constructs immersions of the twice-punctured Riemann sphere into R^3.
result Family of constant mean curvature surfaces constructed.
Alternative proofs for various inequalities on Riemannian manifolds.
problem Various functional inequalities on Riemannian manifolds.
method Generic functional inequality, Riccati pairs, solving Riccati-type ODE.
result Alternative proofs for multiple inequalities, including Hardy-type and Caccioppoli inequalities.
Study phase transitions in noisy transformer dynamics on spheres.
problem Understanding phase transitions in noisy transformer dynamics on spheres.
method Sharp Beckner--Onofri/logarithmic HLS inequality, Funk--Hecke/Bessel coefficients, degree-two quartic obstruction.
result Sharp global-minimizer dichotomy and phase transitions in noisy transformer dynamics in arbitrary dimension.
The tetrahedral index connects to a q-Bessel function, revealing new mathematical techniques.
problem Exploring connections between the tetrahedral index and Hahn-Exton q-Bessel function.
method Establishing a correspondence between the tetrahedral index and the q-Bessel function.
result New techniques and conjectures in q-hypergeometric theory.
We consider the exact path sampling of the squared Bessel process and some other continuous-time Markov processes, such as the CIR model, constant elasticity of variance diffusion model, and hypergeometric diffusions, which can all be obtained from a squared Bessel process by using a change of variable, time and scale …
Directly simulates squared Bessel processes efficiently.
problem Simulating squared Bessel processes accurately and efficiently.
method Two-dimensional Chebyshev expansion for non-central chi-square distribution inverse.
result Accurate and efficient simulation for various degrees of freedom.
We study the geometry and partial differential equations arising from the consideration of Frobenius determinants, also called-group-determinants. This leads us to address some aspects of twistor theory as well as some extensions of Bessel functions.
The paper examines eigenvalues and inequalities on Riemannian manifolds.
problem Eigenvalue behavior and volume growth on Riemannian manifolds.
method Analyzes the asymptotic behavior of the first eigenvalues of balls on Riemannian manifolds.
result Sharp estimates of volume growth and Hardy inequalities under spectral conditions.
New simulation method simplifies Heston model with Poisson conditioning for better accuracy and efficiency.
problem Computational expense in exact simulation schemes for Heston model.
method Proposes a new exact simulation scheme without modified Bessel function evaluations, leveraging conditional integrated variance simplification.
result Good performance in terms of accuracy, efficiency, and reliability compared to existing methods.
Paper analyzes multidimensional PIDEs for financial modeling, proving existence and uniqueness in Bessel spaces.
problem Analyzing solutions of non-local nonlinear PIDEs in multidimensional spaces.
method Employing abstract semilinear parabolic equations theory in Bessel potential spaces.
result Existence and uniqueness of solutions for a wide class of Lévy measures in multidimensional spaces.
Following Donaldson's oppenness theorem on deforming a conical Kähler-Einstein metric, we prove a parabolic Schauder-type estimate with respect to conical metrics. As a corollary, we show that the conical Kähler-Ricci Flow exists for short time. The key is to establish the relevant heat kernel estimates, where we use t…
This paper establishes inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane.
problem Establishing higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane.
method Developing factorization theorems and introducing Geller's operators, combining with Helgason-Fourier analysis and kernel estimates.
result Established higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane.
Paper proves existence and uniqueness of solutions to PIDEs in Bessel spaces for option pricing.
problem Existence and uniqueness of solutions to PIDEs in Bessel spaces.
method Abstract semilinear parabolic equations and Bessel potential spaces.
result Proves existence and uniqueness of solutions in Bessel potential spaces.
Global harmonic maps into SU(1,1) constructed from Smyth potentials using DPW method.
problem Globality of harmonic maps constructed from Smyth potentials in SU(1,1).
method Construct harmonic maps into SU(1,1) using the DPW method, solving a Riemann-Hilbert problem to achieve global Iwasawa factorization.
result Globality of the constructed harmonic maps proved using Bessel functions and asymptotic expansions.
New isoperimetric inequality for clamped plates in RCD(0,N) spaces, sharp and stable.
problem Fine properties of the principal frequency of clamped plates in RCD(0,N) spaces.
method Analyzing the RCD(0,N) spaces and applying isoperimetric inequalities.
result Sharp isoperimetric inequality for the principal frequency of clamped plates in RCD(0,N) spaces.
This paper develops a novel analytically tractable Neumann series of Bessel functions representation for pricing (and hedging) European-style double barrier knock-out options, which can be applied to the whole class of one-dimensional time-homogeneous diffusions even for the cases where the corresponding transition den…
This paper extends barrier option pricing to CIR and CEV models using semi-closed form solutions.
problem Pricing barrier options in time-dependent CEV and CIR models.
method Developed two new methods: Bessel potentials and generalized integral transform, both applied to Bessel processes.
result The methods provide more accurate and stable pricing compared to finite difference methods, especially for small and large maturities.
This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the so-called Asian options. Starting with research of Yor's in 1992, these questions about exponential functionals of Brownian motion have been studied in terms of …
In sparse Bayesian learning (SBL), Gaussian scale mixtures (GSMs) have been used to model sparsity-inducing priors that realize a class of concave penalty functions for the regression task in real-valued signal models. Motivated by the relative scarcity of formal tools for SBL in complex-valued models, this paper propo…
Proposes a new birth-death process for better modeling of population dynamics.
problem Models of population or opinion dynamics with spurious long-range memory.
method Introduces Bessel-like birth-death process to address the spurious long-range memory.
result Derives equations for the burst and inter-burst duration of the new process.
Paper solves PDEs for optimal investment strategies in volatile markets.
problem Finding optimal investment strategies in volatile markets.
method Numerical methods using time-changed Bessel bridges.
result Solves PDEs for relative arbitrage opportunities in volatility-stabilized markets.
New method for pricing barrier options in time-dependent λ-SABR model.
problem Pricing barrier options in the time-dependent λ-SABR model.
method Modified integral transform method and Fourier-Bessel series solution.
result Semi-analytical solution for barrier options in λ-SABR model.
This work extends Tweedie's formulae to non-Gaussian processes for better diffusion model generation.
problem Limited exploration of non-Gaussian diffusion models and corresponding Tweedie's formulae.
method Extended Tweedie's formulae to geometric Brownian motion, squared Bessel, and Cox-Ingersoll-Ross processes.
result Demonstrated potential of non-Gaussian models in image and financial time series generation.
This article concerns new off-diagonal estimates on the remainder and its derivatives in the pointwise Weyl law on a compact n-dimensional Riemannian manifold. As an application, we prove that near any non self-focal point, the scaling limit of the spectral projector of the Laplacian onto frequency windows of constant …
Solves a long-standing problem on step-two groups with exact formulas.
problem Long-standing Gaveau--Brockett open problem on step-two groups.
method Combining Varadhan's formulas, heat kernel, and operator convexity.
result Exact formula for Carnot--Carathéodory distance on step-two groups.
It is shown that most of the well-known basic results for Sobolev-Slobodeckii and Bessel potential spaces, known to hold on bounded smooth domains in Rn, continue to be valid on a wide class of Riemannian manifolds with singularities and boundary, provided suitable weights, which reflect the nature of the s…
Model predicts stock market volatility, leading to successful trading.
problem Real-time risk management in stock markets.
method Algebraic theory of news impact, Bessel and hypergeometric functions, ML procedures.
result Trading system proved successful in historical and real-time experiments.
We propose two main applications of Gyöngy (1986)'s construction of inhomogeneous Markovian stochastic differential equations that mimick the one-dimensional marginals of continuous Itô processes. Firstly, we prove Dupire (1994) and Derman and Kani (1994)'s result. We then present Bessel-based stochastic volatility mod…
Model financial market with fundraiser and stock, derive option prices.
problem Derive option prices in a market with a fundraiser and multiple solutions to the Black-Scholes equation.
method Model financial market with two types of agents, use Pitman's theorem for Bessel process, derive option prices using numerical scheme.
result Derive option prices for European options and call options in a market with a bubble.
In this paper we analytically study the problem of pricing an arithmetically averaged Asian option in the path integral formalism. By a trick about the Dirac delta function, the measure of the path integral is defined by an effective action functional whose potential term is an exponential function. This path integral …
The paper examines how the angle between inputs in ReLU networks decreases with depth, impacting training.
problem Depth degeneracy in neural networks, leading to constant function behavior on initialization.
method Combinatorial expansions and Monte Carlo experiments to analyze the angle between inputs in ReLU networks of increasing depth.
result The angle between inputs in ReLU networks decreases exponentially with depth, leading to constant function behavior on initialization.
Paper analyzes EM algorithm's trajectory in 2MLR, revealing cycloid behavior.
problem Understanding the convergence and trajectory of EM algorithm in 2MLR.
method Explicit closed-form expressions for EM updates, recurrence relation derivation at population level.
result EM iterations lie on a cycloid trajectory, leading to theoretical estimate of convergence exponent.
Unique solutions found for diffusive martingale problems.
problem Finding unique solutions to Cauchy problems for diffusive real-valued strict local martingales.
method Provided sets of smooth functions under local Hölder and Engelbert-Schmidt conditions for unique classical and weak solutions.
result Unique solutions found for specific martingale models.
We investigate large changes, bursts, of the continuous stochastic signals, when the exponent of multiplicativity is higher than one. Earlier we have proposed a general nonlinear stochastic model which can be transformed into Bessel process with known first hitting (first passage) time statistics. Using these results w…
The paper defines function spaces on manifolds with bounded or singular geometries.
problem Defining function spaces on manifolds with various geometries.
method Introduces and analyzes Sobolev, Besov, and Bessel potential spaces on uniformly regular and singular Riemannian manifolds.
result Demonstrates maximal regularity for a linear parabolic problem on singular manifolds.
A rather complete investigation of anisotropic Bessel potential, Besov, and Hölder spaces on cylinders over (possibly) noncompact Riemannian manifolds with boundary is carried out. The geometry of the underlying manifold near its 'ends' is determined by a singularity function which leads naturally to the study of weigh…
The paper studies projections of asset prices under equivalent martingale measures.
problem Understanding the impact of information on asset price bubbles and arbitrage opportunities.
method Analyzes optional projections of local martingales into a smaller filtration under equivalent martingale measures.
result Provides general results and specific examples like inverse Bessel process and stochastic volatility models.
Extensions of Brownian motion to singular surfaces are studied.
problem Diffusion across singularities on surfaces.
method One-parameter family of Grushin-type singularities, heat crossing analysis, isometry group respect, Bessel processes.
result Complete description and classification of diffusions for various singularity cases.
In this paper, we present a Bayesian channel estimation algorithm for multicarrier receivers based on pilot symbol observations. The inherent sparse nature of wireless multipath channels is exploited by modeling the prior distribution of multipath components' gains with a hierarchical representation of the Bessel K pro…
Strict local martingales may admit arbitrage opportunities with respect to the class of simple trading strategies. (Since there is no possibility of using doubling strategies in this framework, the losses are not assumed to be bounded from below.) We show that for a class of non-negative strict local martingales, the s…
The paper introduces new methods for Asian option pricing using Laguerre quadrature.
problem Developing accurate pricing models for Asian options.
method Utilizes Laguerre quadrature and diffusion kernel approach.
result Demonstrates new techniques to solve complex Asian option pricing equations.
We provide an affirmative answer to a question posed by Tod \cite{Tod:1995b}, and construct all four-dimensional Kahler metrics with vanishing scalar curvature which are invariant under the conformal action of Bianchi V group. The construction is based on the combination of twistor theory and the isomonodromic problem …