This paper proves properties of convex integrands and their duals.
problem Properties of convex integrands and their duals.
method Analyzing and stable convex integrands.
result Dual convex integrands of stable convex ones are also stable.
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Simultaneous smoothness and simultaneous stability of a strictly convex integrand and its dualmath.GT
The paper examines smoothness and stability of convex integrands and their duals.
problem Properties of convex integrands and their duals.
method Investigation of simultaneous smoothness and stability of a strictly convex integrand and its dual.
result For a strictly convex integrand, its dual is of class if and only if the integrand is strictly convex.
Dual representation of Kantorovich functional using martingale measures.
problem Representation of Kantorovich functional on Skorokhod space.
method Choquet capacity generated by martingale measures with constraints.
result Dual representation of Kantorovich functional.
Chiral string integrands simplify to ambitwistor string integrands in the tensionless limit.
problem Understanding the relationship between chiral and ambitwistor string integrands.
method Analyzing the tensionless limit of chiral superstring integrands.
result Chiral superstring integrands reduce to ambitwistor string integrands in the tensionless limit.
In this paper we study a risk-minimizing hedging problem for a semimartingale incomplete financial market where d+1 assets are traded continuously and whose price is expressed in units of the numéraire portfolio. According to the so-called benchmark approach, we investigate the (benchmarked) risk-minimizing strategy in…
Stable convex integrands are open and dense in smooth convex integrands.
problem Characterizing stability of convex integrands.
method Whitney topology analysis.
result Set of stable convex integrands is open and dense.
Kernel quadratures can be consistent even when the integrand is less smooth than assumed.
problem Kernel quadratures assume smoothness of integrands, but this assumption is often violated in practice.
method Derives convergence rates for kernel quadratures in misspecified settings, relating them to the lesser smoothness of the integrand.
result Kernel quadratures can be consistent even when the integrand is less smooth than assumed, providing alternatives to Monte Carlo integration.
Quantum speedup for Monte Carlo integration reduces integrand calls.
problem Reducing the number of calls to the integrand subroutine in high-dimensional Monte Carlo integration.
method Combining nested quantum amplitude estimation with pseudorandom numbers for separable integrands.
result Significant reduction in the number of integrand calls for high-dimensional integration.
Ambitwistor string matches superstring chiral integrands at zero tension.
problem Matching scattering amplitudes in superstring theory and ambitwistor string theory.
method Direct computation and reduction to ordinary moduli space.
result Chiral half integrands of superstring match those of ambitwistor string in the zero tension limit.
In anomaly-free quantum field theories the integrand in the bosonic functional integral--the exponential of the effective action after integrating out fermions--is often defined only up to a phase without an additional choice. We term this choice ``setting the quantum integrand''. In the low-energy approximation to M-t…
New geometric interpretations reveal structure of AC integrands.
problem Understanding and verifying the atomic condition for integrands.
method Reinterpretation of atomic condition in convex geometry.
result Quantitative versions EC and QEC proposed; stability and regularity results.
In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class . Moreover, applications of this result are given.
Bayesian optimization for expensive integrands achieves optimal performance.
problem Optimizing functions with expensive integrands in noisy conditions.
method Bayesian optimization with discretization-free value of information optimization.
result Achieves optimal performance in noisy and smooth conditions.
Extends Euler calculus to continuous integrands using curvature.
problem Limitations of Euler calculus with simple functions.
method Integrates with respect to Gaussian curvature within O-minimal theories.
result Satisfies a Fubini theorem and extends to a functor.
Novel approach for estimating conditional expectations using Bayesian quadrature.
problem Estimating conditional expectations with costly evaluations.
method Probabilistic numerical methods incorporating prior smoothness knowledge.
result Fast convergence rate and uncertainty quantification.
In this paper we consider the evolution of a graph-like hypersurface by anisotropic mean curvature flow, under some restrictions on the anisotropic area integrand. We find interior estimates (in both time and space) on the gradient of such hypersurfaces, depending only on the height of the graph and the anisotropic are…
The goal of the present paper is to investigate the algebraic structure of global conformal invariants of submanifolds. These are defined to be conformally invariant integrals of geometric scalars of the tangent and normal bundle. A famous example of a global conformal invariant is the Willmore energy of a surface. In …
Counterexample shows Ito integrand needn't be locally square integrable.
problem Ito integrand's square integrability condition is not always met.
method Provided a counterexample to Ito's Lemma's integrability condition.
result Ito integrand needn't be locally square integrable.
This paper forms part of a larger work where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of "global conformal invariants"; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a…
Quasi-Monte Carlo with Domain Transformation for Efficient Fourier Pricing of Multi-Asset Optionsq-fin.CP
This work improves Fourier pricing for multi-asset options using RQMC with domain transformation.
problem Efficiently pricing multi-asset options in high dimensions with Fourier methods.
method Randomized quasi-Monte Carlo (RQMC) with domain transformation to handle singularities.
result RQMC with domain transformation provides accurate and scalable Fourier pricing for multi-asset options.
Simplifies Wulff theorem for crystalline shapes using Minkowski Theory.
problem Proving the Wulff theorem for crystalline integrands.
method Direct approach using Minkowski Theory to exploit convex properties.
result Simpler proof of the Wulff theorem for crystalline shapes.
Study proves uniform regularity for surface energies, critical and subcritical.
problem Establishing regularity for surface energies in critical and subcritical cases.
method Uniform ε-regularity estimates for intrinsic elliptic Lagrangians.
result Critical points of surface energies are uniformly regular for a wide class of Lagrangians.
This is the second in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed …
Smooths basket option pricing for faster computation.
problem Efficiently pricing basket options with non-smooth integrands.
method Mollify the payoff function using exact conditional expectation with smooth integrand.
result High-order method performs significantly faster than Monte Carlo or Quasi Monte Carlo.
On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed into a Chern polynomial (the integrand of a Chern number) and divergences of one …
This is the fifth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed a…
The study proves that certain minimal surfaces are flat under specific conditions.
problem Characterizing minimal surfaces in anisotropic spaces.
method Proving a Bernstein theorem for -anisotropic minimal hypersurfaces.
result The only entire smooth solutions to the -anisotropic minimal hypersurfaces equation are linear functions.
Proves energy expression on Poincaré-Einstein spaces.
problem Computing renormalized Yang-Mills energy on Poincaré-Einstein manifolds.
method Generalizes Chang-Qing-Yang method for renormalized volumes and uses scattering theory for Schrödinger operators.
result Agrees with anomaly boundary integrand in seven dimensions.
Constructs polyhedral chains with prescribed tangent plane distributions.
problem Constructing polyhedral chains with specific tangent plane distributions.
method Explicit construction of polyhedral chains that approximate prescribed measures on Grassmannian.
result Polyconvexity is equivalent to quasiconvexity of associated Q-integrands under certain conditions.
Polyconvex energies with conformal invariance have smooth stationary points outside a discrete set.
problem Stationary points of conformally invariant polyconvex energies
method Proving smoothness of stationary points
result Smooth stationary points outside a discrete set
In 1963, K.P.Grotemeyer proved an interesting variant of the Gauss-Bonnet Theorem. Let M be an oriented closed surface in the Euclidean space R^3 with Euler characteristic χ(M), Gauss curvature G and unit normal vector field n. Grotemeyer's identity replaces the Gauss-Bonnet integrand G by the normal moment <a,n>^2G, w…
Paper defines a pathwise Ito integral without probabilistic assumptions.
problem Defines a non-probabilistic Ito integral for non-stochastic processes.
method Provides constructions for the pathwise Ito integral under various conditions.
result Existence of the integral for cadlag integrands and integrators with bounded jumps.
This is the first in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global confor- mal invariants"; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed …
The decomposition of global conformal invariants IV: A proposition on local Riemannian invariantsmath.DG
This is the fourth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed …
This is the last in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as…
The Gauss-Bonnet curvature of order is a generalization to higher dimensions of the Gauss-Bonnet integrand in dimension , as the usual scalar curvature generalizes the two dimensional Gauss-Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals…
NCV uses neural networks to improve Monte Carlo integration.
problem Improving variance reduction in parametric Monte Carlo integration.
method NCV combines a normalizing flow and a neural network to approximate the integrand and solve the integral equation, with a neural importance sampler to estimate the difference.
result NCV achieves state-of-the-art performance in light transport simulation with reduced noise and negligible bias.
Study on error rates for approximating rough volatility models.
problem Simulation of rough volatility models with fractional Brownian motion.
method Analysis of weak error rates for numerical schemes, focusing on fBm and cubic test functions.
result Convergence rates for approximations are for exact left-point discretization and for hybrid schemes.
The paper analyzes kernel-based quadrature in misspecified settings, providing convergence rates and robustness conditions.
problem Analyzing kernel-based quadrature in settings where the test integrand is less smooth than the RKHS.
method Convergence analysis based on two assumptions: constant weights or minimum distance between design points.
result Derives convergence rates and conditions for robustness in Bayesian quadrature under misspecification.
Local index theorem for chiral geometric operators proved using heat kernel.
problem Proving a local index theorem for geometric first-order differential operators.
method Using Gilkey's invariance theory and heat kernel techniques.
result Supertrace of heat kernel converges to Chern-Weil form.
On the Gauss Map of Anisotropic Minimal Surfaces and applications to the Morse Index estimatesmath.DG
Study on Gauss map of anisotropic minimal surfaces with Morse index estimates.
problem Estimating the Morse index of anisotropic minimal surfaces.
method Local analysis of Gauss map, conformal geometric techniques applied to the Gauss map.
result Upper and lower estimates for the Morse index of anisotropic minimal surfaces.
Bayesian Quadrature speeds up integration by selecting batches of points instead of single points.
problem Efficiently parallelizing Bayesian Quadrature for integration over non-negative integrands.
method Developed methods to select batches of points at each step, based on recent batch Bayesian Optimization.
result Significantly reduces computation time, especially for expensive integrands.
We study underlying geometric structures for integral variational functionals, depending on submanifolds of a given manifold. Applications include (first order) variational functionals of Finsler and areal geometries with integrand the Hilbert 1-form, and admit immediate extensions to higher-order functionals.
A new method optimizes Fourier pricing for multi-asset options using adaptive quadrature.
problem Efficiently pricing multi-asset options in Lévy models.
method Optimized damping parameters and hierarchical adaptive quadrature.
result Significant speed-up in computational time for up to six dimensions.
Study market models without concavity assumptions, proving representability and extending no-arbitrage concepts.
problem Market models without concavity assumptions.
method Defining market models, proving representability, extending no-arbitrage concepts.
result Market models can be represented as normal integrands under sequential upper-semicontinuity.
Foundation for robust finance using rough path theory.
problem Mathematical models of financial markets under Knightian uncertainty.
method Introducing Property (RIE) for càdlàg paths, proving existence of rough integrals, verifying admissibility of trading strategies.
result Existence and stability of rough path integrals for non-gradient integrands.
New theorem handles stochastic Volterra semimartingales.
problem Classic Fubini theorem restrictions for Volterra semimartingales.
method Introduced measure-valued stochastic integration.
result Proved new stochastic Fubini theorem.
Tensor approach simplifies Euclidean space descriptions.
problem Simplifying tensor descriptions of Euclidean spaces.
method Emphasizes geometric vectors in tensor description.
result Proved integral identities with vector integrands.