Optimizes threshold selection for variance estimation in financial models.
problem Estimating integrated variance in financial models with jumps.
method Optimizes threshold selection using mean and conditional mean square error criteria.
result Proposes a novel method to approximate the optimal threshold.
Paper proves uniform continuity bounds for complex Monge-Ampère solutions.
problem Estimating the continuity of solutions to complex Monge-Ampère equations.
method PDE-based approach from fully non-linear equations in Kähler geometry.
result Uniform and sharp estimate for the modulus of continuity.
Study on existence and properties of continuous solutions to complex Hessian equations.
problem Existence and properties of continuous solutions to complex Hessian equations.
method Established new capacity estimates and weak stability estimates for the m-Hessian measure. result Existence of continuous solutions to the complex Hessian equation under certain conditions.
Study on Hölder continuity of complex Monge-Ampère solutions on Stein spaces.
problem Understanding continuity of solutions to complex Monge-Ampère equations on Stein spaces.
method Analyzing solutions with Lp densities and Hölder boundary data on Stein spaces with isolated singularities. result Solutions are Hölder continuous outside singular points if boundary data is Hölder continuous.
Proves conditions for Fourier transforms in rank 1 symmetric spaces.
problem Understanding Fourier transform bounds in symmetric spaces.
method Proves sufficient and necessary conditions using Lipschitz and Fourier type integral conditions.
result Establishes bounds for Fourier transforms in rank 1 symmetric spaces with specific moduli of continuity.
We discuss the maximum modulus principle, and weak unique continuation, for CR functions on an abstract almost CR manifold M. We investigate these matters under the assumption of weak pseudoconcavity, and obtain sharp results about propagation along Sussmann leaves.
We derive sharp estimates on modulus of continuity for solutions of the heat equation on a compact Riemannian manifold with a Ricci curvature bound, in terms of initial oscillation and elapsed time. As an application, we give an easy proof of the optimal lower bound on the first eigenvalue of the Laplacian on such a ma…
We continue the study of the variation of the p--modulus of a foliation initiated by the first author. We derive the formula for the second variation which allows to study p--stable foliations. We obtain some results concerning codimension one p--stable foliations. Moreover, we derive the equation for the critica…
Researchers establish bounds and continuity of decomposed Möbius energies using cosine formula.
problem Estimating the bounds and continuity of decomposed Möbius energies.
method Using the cosine formula to evaluate upper and lower bounds and modulus of continuity of decomposed energies.
result Affirmative answer to the question of estimating decomposed energies using the cosine formula.
New bounds on the wildness of Bing's involution.
problem Analyzing the wildness of Bing's involution in terms of its modulus of continuity.
method Proving a nearly exponential modulus of continuity for topologically conjugate involutions.
result The modulus of continuity for topologically conjugate Bing involutions is at least exponential up to a polylogarithmic factor.
We consider the optimal stopping problem $v^{(\eps)}:=\sup_{τ\in\mathcal{T}_{0,T}}\mathbb{E}B_{(τ-\eps)^+}$ posed by Shiryaev at the International Conference on Advanced Stochastic Optimization Problems organized by the Steklov Institute of Mathematics in September 2012. Here T>0 is a fixed time horizon, $(B_t)_{0\le…
Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.
problem Understanding moduli of continuity for fully nonlinear parabolic equations.
method Proving moduli of continuity of viscosity solutions are subsolutions of one-dimensional parabolic equations.
result Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations with bounded initial data.
We develop methods to efficiently approximate data in metric spaces without additional assumptions.
problem Efficiently approximating data in metric spaces without imposing structural assumptions.
method Identify discrete modulus of continuity, investigate consistency, propose algorithm, and develop approximation theory.
result Consistent approximation of data in metric spaces without structural assumptions.
New recursive relations for symmetries of special 2-flags discovered.
problem Local finite classification and continuous numerical modulus of special 2-flags.
method Recursive relations derived for all infinitesimal symmetries of special 2-flags.
result Existence of a continuous numerical modulus of special 2-flags in length seven.
Proposes DMOC for more nuanced neural network robustness.
problem Lipschitz continuity is too coarse for nuanced data-dependent behavior.
method Data-driven, architecture-agnostic framework based on DMOC.
result DMOC provides a finer notion of robustness relative to data distribution.
Sharp estimates derived for quasilinear equations on metric measure spaces.
problem Estimating solutions and eigenvalues of quasilinear equations on smooth metric measure spaces.
method Sharp estimates derived using comparisons with one-dimensional equations.
result Optimal lower bounds for the first Dirichlet eigenvalue of quasilinear operators.
Researchers develop neural networks for approximating functions in Banach spaces.
problem Approximating Banach space valued continuous functions.
method Quasi-interpolation Banach space valued neural network operators using algebraic sigmoid functions.
result Jackson type inequalities for function approximation.
We establish the estimates of modulus of continuity for viscosity solutions of nonlinear evolution equations on manifolds, extending previous work of B. Andrews and J. Clutterbuck for regular solutions on manifolds \cite{AC3} and the first author's recent work for viscosity solutions in Euclidean spaces \cite{me1}.
Unified approach for sample aggregation in transfer learning across various divergence measures.
problem Optimizing sample aggregation from source to target distributions for improved target performance.
method Unified algorithmic approach that adapts to multiple divergence measures via a weak modulus of transfer.
result Unified approach achieves near optimal rates in terms of the unknown strong modulus, applicable in more general settings.
Motivated by the pricing of lookback options in exponential Lévy models, we study the difference between the continuous and discrete supremum of Lévy processes. In particular, we extend the results of Broadie et al. (1999) to jump-diffusion models. We also derive bounds for general exponential Lévy models.
In this paper, we investigate the moduli of continuity for viscosity solutions of a wide class of nonsingular quasilinear evolution equations and also for the level set mean curvature flow, which is an example of singular degenerate equations. We prove that the modulus of continuity is a viscosity subsolution of some o…
Study develops time-continuous models and probabilistic descriptions for agent-based economic market models.
problem Formulating and describing agent-based economic market models in a time-continuous and probabilistic manner.
method Derived time-continuous formulations, discussed impact of time-scaling, proved stability, presented probabilistic descriptions using kinetic theory.
result Time-continuous formulations and probabilistic descriptions for agent-based economic market models.
Study the discontinuity of functions not embeddable in Euclidean space.
problem Understanding discontinuity of non-embeddable functions.
method Define a modulus of discontinuity and establish lower bounds.
result Quantified nonembeddability results and topological Tverberg theorem.
We extend the traditional worst-case, minimax analysis of stochastic convex optimization by introducing a localized form of minimax complexity for individual functions. Our main result gives function-specific lower and upper bounds on the number of stochastic subgradient evaluations needed to optimize either the functi…
Minimal width din+1 allows ReLU nets to approximate any continuous function of din variables.
problem Approximating continuous functions using ReLU nets with minimal width.
method Analyzing the expressive power of depth in neural nets with ReLU activations.
result Minimal width din+1 is necessary and sufficient for ReLU nets to approximate any continuous function of din variables. Universal approximation for rough paths and Lévy processes.
problem Approximating continuous functionals of càdlàg paths.
method Linear functionals of time-extended signatures.
result Universal approximation theorem for continuous functionals of càdlàg paths.
Let C be a subset of Rn (not necessarily convex), f:C→R be a function, and G:C→Rn be a uniformly continuous function, with modulus of continuity ω. We provide a necessary and sufficient condition on f, G for the existence of a convex function F∈C1,ω(Rn) …
Study on harmonic functions in RCD spaces, focusing on singular points and vanishing gradients.
problem Behavior of harmonic functions at singular points of RCD spaces.
method Analysis of tangent cones and modulus of continuity.
result Gradient of harmonic functions vanishes at certain singular points.
A theorem controls the relationship between dimensions of continua.
problem Understanding the relationship between different dimensions of continua.
method Introduced a controlled version of the Hahn-Mazurkiewicz Theorem.
result Established a relationship between SDim(X) and HDim(X). Establishes a lower bound for Kähler hyperbolicity modulus in hyperconvex domains and bounded strongly pseudoconvex domains.
problem Kähler hyperbolicity modulus for simply-connected Kähler hyperbolic manifolds
method Computes the Kähler hyperbolicity modulus for bounded symmetric domains
result Establishes a lower bound for the Kähler hyperbolicity modulus in terms of the boundary behavior of the gradient length of a plurisubharmonic function
Curve shortening flow increases annulus modulus.
problem Behavior of annulus modulus under curve shortening flow.
method Nested curves evolving under curve shortening flow.
result Modulus of enclosed annulus is monotonically increasing.
Consider power utility maximization of terminal wealth in a 1-dimensional continuous-time exponential Levy model with finite time horizon. We discretize the model by restricting portfolio adjustments to an equidistant discrete time grid. Under minimal assumptions we prove convergence of the optimal discrete-time strate…
Study negative discount rate effects on perpetual options in Lévy models.
problem Negative discount rate impacts perpetual American and Swing options in Lévy models.
method Analyze perpetual American and put options in exponential Lévy models with negative discount rate, identify critical continuation prices, and generalize to Swing type problems.
result Double continuation region arises in negative discount rate cases, identified by critical prices.
Study of bandit problem with Poisson decision times and Lévy processes.
problem Continuous-time multi-armed bandit problem with Poisson decision times.
method Gittins index policy applied to spectrally one-sided Lévy processes.
result Gittins index converges to classical Lévy bandit index.
We study non-variational degenerate elliptic equations with high order singular structures. No boundary data are imposed and singularities occur along an {\it a priori} unknown interior region. We prove that positive solutions have a universal modulus of continuity that does not depend on their infimum value. We furthe…
A scattering transform defines a signal representation which is invariant to translations and Lipschitz continuous relatively to deformations. It is implemented with a non-linear convolution network that iterates over wavelet and modulus operators. Lipschitz continuity locally linearizes deformations. Complex classes o…
Sharp lower bounds for eigenvalues on weighted p-Laplacian manifolds.
problem Estimating the first nonzero eigenvalue of the weighted p-Laplacian on compact manifolds.
method Sharp gradient comparison theorem and modulus of continuity estimates.
result Proves sharp lower bound estimates for the first nonzero eigenvalue.
Kernel for Lévy rough paths derived from PDE system.
problem Computing similarity measures for Lévy rough paths.
method Developed a PDE system for the expected signature of inhomogeneous Lévy processes.
result Gaussian martingales' expected signature kernel satisfies a Goursat PDE.
Recently, Andrews and Clutterbuck [AC13] gave a new proof of the optimal lower eigenvalue bound on manifolds via modulus of continuity for solutions of the heat equation. In this short note, we give an alternative proof of Theorem 2 in [AC13]. More precisely, following Ni's method ([Ni13, Section 6]) we give an ellipti…
The p--modulus modp(F) of a foliation F on a Riemannian manifold M is a generalization of extremal length of plane curves introduced by L. Ahlfors. We study the variation t↦modp(Ft) of the modulus. In particular, we consider product of moduli of orthogonal fo…
In the product space H^n \times R; we obtain uniform a priori C^0 horizontal length estimates, uniform a priori C^1 boundary gradient estimates, as well as uniform modulus of continuity, for a class of horizontal minimal equations. In two independent variables, we derive a certain uniform global a priori C^1 estimates …
New neural network architecture with height adds expressive power.
problem Expressiveness of neural networks limited by width and depth.
method Introduces height as a new hyper-parameter in neural network architecture.
result Neural networks with height achieve significantly better approximation of functions.
Three-hidden-layer neural networks can approximate Hölder continuous functions uniformly with exponential rate.
problem Approximating Hölder continuous functions with neural networks.
method Introduced Floor-Exponential-Step (FLES) networks with three hidden layers.
result Uniform approximation of Hölder continuous functions with an exponential rate.
This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli-Chern class on compact complex manifolds, and proved that the (1,1) curvature form of the Levi-Civita connection represents the first Aeppli-Chern class which is a natural link between Riemannian geometry and complex geometry. …
New network approximates functions with error decreasing with network width and depth.
problem Approximating functions with high-dimensional data.
method Floor-ReLU networks with specific width and depth.
result Approximation error decreases as network width and depth increase.
In this paper we propose a general derivative pricing framework which employs decoupled time-changed (DTC) Lévy processes to model the underlying asset of contingent claims. A DTC Lévy process is a generalized time-changed Lévy process whose continuous and pure jump parts are allowed to follow separate random time scal…
The paper defines and analyzes Poissonian occupation times for negative Lévy processes.
problem Analyzing the time spent below zero for Lévy processes with interruptions.
method Introduces Poissonian occupation times for spectrally negative Lévy processes.
result Extends results on continuous observation to interrupted observation.
Study on regularity of exceptional actions and moduli of continuity for circle diffeomorphisms.
problem Regularity of exceptional actions of groups on the circle.
method Analysis of C1,α diffeomorphisms with free orbits and bounded derivatives. result Existence of exceptional C1,α diffeomorphisms under certain conditions on α.