Study examines explainable machine learning for monotonic models, finding Integrated gradients better for strong monotonicity.
problem Applying explainable machine learning to science-informed models.
method Proposed axioms for monotonicity, tested Shapley value and Integrated gradients methods.
result Integrated gradients provides better explanations for strong monotonicity.
The paper studies frequency monotonicity for solutions of nonlinear equations under Ricci flow.
problem Frequency monotonicity for positive solutions of nonlinear equations under Ricci flow.
method Obtained parabolic frequency monotonicity for solutions of two nonlinear parabolic equations with bounded Ricci curvature.
result Established integral type Harnack inequalities using parabolic frequency monotonicity.
In this paper we prove a monotonicity formula for the integral of the mean curvature for complete and proper hypersurfaces of the hyperbolic space and, as consequences, we obtain a lower bound for the integral of the mean curvature and that the integral of the mean curvature is infinity.
Developed a monotone numerical method for MV portfolio optimization under jump-diffusion models.
problem Efficiently optimizing portfolios with jump-diffusion dynamics and investment constraints.
method Strictly monotone numerical integration method using Fourier transforms and composite quadrature rules.
result Proven to be ℓ∞-stable and pointwise consistent, converging to the MV optimization solution. Study solves optimal portfolio selection using HJB equation.
problem Optimal portfolio selection problem.
method Maximal monotone operator method, Banach fixed-point theorem, Fourier transform, monotone operators technique.
result Existence and uniqueness of solution to HJB equation.
The problem of existence of arbitrage free and monotone CDO term structure models is studied. Conditions for positivity and monotonicity of the corresponding Heath-Jarrow-Morton-Musiela equation for the x-forward rates with the use of the Milian type result are formulated. Two state spaces are taken into account - of…
We give a family of monotone quantities along smooth solutions to the inverse curvature flows in Euclidean spaces. We also derive a related geometric inequality for closed hypersurfaces with positive k-th mean curvature.
New formulas limit minimal submanifolds' area in curved spaces.
problem Bounding minimal submanifolds' area in curved spaces.
method Developed new monotonicity formulae involving energy-like integrals over non-geodesic sets.
result Imply sharp area bounds for minimal submanifolds through a prescribed point.
Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit …
A new method solves complex financial equations efficiently.
problem Solving worst-case and best-case prices for two-factor uncertain volatility models.
method Decompose and integrate, then optimize; piecewise constant control; closed-form Green's functions; 2D convolution integrals; monotone numerical integration; Fast Fourier Transforms.
result The method efficiently computes the value function and optimal control, converging to the viscosity solution of the HJB equation.
New method for optimizing risk in financial models using Fourier transforms.
problem Optimizing risk in financial models with multi-period mean-CVaR.
method Strictly monotone 2D integration scheme via Fourier-trained transition kernels.
result Established robust and accurate optimization method for financial models.
This paper develops a novel numerical method for pricing American options in a two-asset jump-diffusion model.
problem Pricing American options under correlated two-asset jump-diffusion models using finite difference methods often fails to preserve monotonicity and accurately discretize jump integrals.
method Introduces a novel monotone integration scheme to solve 2-D Partial Integro-Differential Equations (PIDEs) efficiently and accurately.
result The proposed method ensures convergence to the viscosity solution of the variational inequality and is both ℓ∞-stable and consistent. Derive monotone quantities for harmonic functions on asymptotically flat 3-manifolds with nonnegative scalar curvature.
problem Derive monotone quantities for harmonic functions on asymptotically flat 3-manifolds with nonnegative scalar curvature.
method Follow the strategy developed in Miao.
result Derive monotone quantities for harmonic functions on asymptotically flat 3-manifolds with nonnegative scalar curvature.
Bayesian optimization with preference learning using monotonic neural networks.
problem Optimizing complex systems with multiple conflicting objectives.
method Proposes a neural network ensemble for utility surrogate modeling, leveraging monotonicity.
result Demonstrates superior performance compared to existing methods.
The paper evaluates the importance of monotonicity in AI fairness across various fields.
problem Ensuring fairness in AI applications across criminology, education, health care, and finance.
method Theoretical reasoning, simulation, and extensive empirical analysis of monotonic neural additive models (MNAMs).
result Monotonicity is essential for fairness in AI ethics and society, especially in criminology, education, health care, and finance.
Motivated by the problem of optimal portfolio liquidation under transient price impact, we study the minimization of energy functionals with completely monotone displacement kernel under an integral constraint. The corresponding minimizers can be characterized by Fredholm integral equations of the second type with cons…
The paper derives inequalities for mean curvatures of hypersurfaces in Riemannian manifolds.
problem Geometric inequalities for mean curvatures of hypersurfaces in Riemannian manifolds.
method Comparison formula via Reilly's identities; geometric inequalities derived.
result Sharp lower bound for total first mean curvature in dimension 3.
Using the monotonicity formulas of Colding and Minicozzi, we prove that on any complete, non-parabolic Riemannian manifold (M3,g) with non-negative Ricci curvature, the asymptotic weighted scaling invariant integral of scalar curvature has an explicit bound in form of asymptotic volume ratio.
The paper studies gradient estimates and monotonicity of parabolic frequency for solutions to the Laplacian G_2 flow.
problem Gradient estimates and monotonicity of parabolic frequency for solutions to the Laplacian G_2 flow.
method Gradient estimates and Harnack inequalities for heat equations under the Laplacian G_2 flow.
result Monotonicity of parabolic frequency and backward uniqueness for positive solutions.
Study on risk measures using distorted Choquet integrals with random distortions.
problem Developing risk measures under random distortions of capacities.
method Introducing and analyzing randomly distorted Choquet integrals with respect to a distorted capacity, establishing properties and providing representations.
result Representation of comonotonic additive conditional risk measures using G-randomly distorted Choquet integrals.
We define an integer graded symplectic Floer cohomology and a spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopies. Such an integer graded Floer cohomology is an integral lifting of the usual Floer-Oh cohomology with $Z_{\Si (L)}$ grading. As one of applications of the sp…
We consider the learning algorithms under general source condition with the polynomial decay of the eigenvalues of the integral operator in vector-valued function setting. We discuss the upper convergence rates of Tikhonov regularizer under general source condition corresponding to increasing monotone index function. T…
Both analytic and geometric forms of an optimal monotone principle for Lp-integral of the Green function of a simply-connected planar domain Ω with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geome…
Characterizes continuity of monotone functionals in mixed topology.
problem Continuity of monotone functionals in mixed topology.
method Characterization through lower semicontinuity and dual representations.
result Continuity in mixed topology is equivalent to dual representation in terms of countably additive measures.
New varifolds with capillary boundary properties studied.
problem Understanding varifolds with specific boundary conditions.
method Introducing a Radon measure on a Grassmannian bundle as a capillary boundary.
result Structural properties, monotonicity inequality, and integral compactness proved.
Develops methods to analyze feature-outcome associations in subpopulations.
problem Challenges in understanding feature-outcome associations in high-dimensional data.
method Geometric decomposition framework using gradient flow and co-monotonicity decomposition.
result Identifies context-dependent patterns and improves statistical power and interpretability.
SurvSurf predicts first hitting times for intermittent events without monotonic violations.
problem Predicting first hitting times for intermittent events with monotonicity guarantees.
method Partially monotonic neural network for sequential events, incorporating unobserved events.
result SurvSurf outperforms existing models in MSE and IBS metrics.
New, shorter proofs for varifolds and flows with improved decay of flatness.
problem Proving regularity theorems for varifolds and flows with bounded first variation and forcing.
method Decay of flatness via weighted monotonicity formulas and viscosity approach.
result Improved proofs with decay of flatness and characterization of blow-ups.
Paper solves overdetermined k-Hessian equation in exterior domains.
problem Overdetermined problem for k-Hessian equation in exterior domains. method Combining integral identities and geometric inequalities, derived general monotone formulas.
result Established general monotone formulas for k-admissible solutions. The paper examines how the first Steklov-Dirichlet eigenvalue changes with the distance between two concentric circles.
problem Investigating the monotonicity of the first Steklov-Dirichlet eigenvalue on eccentric annuli.
method The approach involves showing differentiability, deriving integral expressions for the derivative, and using variational formulations to find upper and lower bounds.
result The paper proves the monotonicity of the first Steklov-Dirichlet eigenvalue on eccentric annuli with respect to the distance between the centers of the inner and outer boundaries.
This paper classifies solutions for a specific geometric problem.
problem Classifying solutions for the planar isotropic Lp dual Minkowski problem. method Converted the ODE for the solution into an integral and studied its asymptotic behavior, duality, and monotonicity.
result Complete classification of solutions for the equation.
Monotonicity of normalized implied-volatility coordinates under no-arbitrage
problem Monotonicity of normalized implied-volatility coordinates under no-arbitrage
method Elementary discrete no-arbitrage proof
result Monotonicity principle extended to Bachelier implied volatility
This paper shows how to learn variational inequalities fast with strong monotonicity.
problem Learning variational inequalities efficiently.
method Extending convex optimization techniques to variational inequalities with strong monotonicity.
result Fast generalization rates of Θ(1/ε) for learning variational inequalities. MCNet improves uncertainty calibration in online advertising by modeling complex relations and balancing performance.
problem Lack of effective calibration for complex relations and context features in online advertising.
method Introduces MCNet with MCF, order-preserving, and field-balance regularizers.
result Superior performance in generating well-calibrated probability predictions on public and industrial datasets.
The paper explores how investors make decisions under disappointment aversion, finding that they prefer not to invest.
problem Continuous-time portfolio selection under generalized disappointment aversion.
method Sufficient and necessary condition for equilibrium strategies via fully nonlinear integral equation.
result Equilibrium strategy under disappointment aversion leads to less investment in the stock market compared to classical utility theory.
In this paper, we obtain a positivity result of a quasi-local mass integral as proposed by Shi and Tam in general dimensions. The main argument is based on the monotonicity of a mass integral in a foliation of quasi-spherical metrics and a positive mass type theorem which was proved by Wang and Yau in the three dimensi…
We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space. We show that integral foliated s…
Sharp lower bound found for integral varifolds' mean curvature.
problem Finding a sharp lower bound for the mean curvature integral of integral varifolds.
method Developed a new approach using integral varifolds and mean curvature.
result A sharp lower bound on the mean curvature integral with critical power for integral varifolds.
Survival regression method improves log-likelihood scores.
problem Improper scoring rules in survival regression models.
method SurvivalMonotonic-net (SuMo-net) with monotonic neural networks.
result SuMo-net achieves state-of-the-art log-likelihood scores.
The paper examines the unexpected losses and risk ratios for co-monotonic alternatives in large portfolios.
problem Understanding the unexpected losses and risk ratios for large portfolios with co-monotonic alternatives.
method Analyzes the asymptotic behavior of unexpected losses and risk ratios for co-monotonic alternatives using monotone cash-additive risk measures and Choquet insurance premia.
result Unexpected losses of large weighted portfolios are of order o(nλn), where λn is the average weight. Formula connects curvature to volume in special geometric spaces.
problem Deriving formulas for curvature in specific geometric spaces.
method Used strong locality of Laplacian and eigenfunction approximation.
result Proved integral type Gauss-Green formula linking curvature to volume.
We study a problem of finding an optimal stopping strategy to liquidate an asset with unknown drift. Taking a Bayesian approach, we model the initial beliefs of an individual about the drift parameter by allowing an arbitrary probability distribution to characterise the uncertainty about the drift parameter. Filtering …
We prove a monotonicity identity for compact surfaces with free boundaries inside the boundary of unit ball in Rn that have square integrable mean curvature. As one consequence we obtain a Li-Yau type inequality in this setting, thereby generalizing results of Oliveira and Soret, and Fraser and Schoen. In th…
This work studies nonnegativity-preserving kernels for stochastic equations and their applications.
problem Nonnegativity preservation in stochastic Volterra equations and related processes.
method Characterization and application of completely monotone kernels; approximation schemes for weak error.
result Positive linear combinations of decaying exponentials can be used for second-order approximation schemes.
Perelman has discovered two integral quantities, the shrinker entropy $\cW$ and the (backward) reduced volume, that are monotone under the Ricci flow $\pa g_{ij}/\pa t=-2R_{ij}$ and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The {\it expanding entr…
Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian distributions, appear naturally in Differential Geometry, Calculus of Variations and Rational Mechanics. The basic differential invariants of a dynamical Lagrangian distribution w.r.t. the action of the group of symplectomorphisms of the a…
Choquet and minimax expectations are equivalent in European option pricing.
problem Pricing European options in incomplete markets.
method Comparing Choquet and minimax expectations in the context of European options.
result Choquet and minimax expectations are equal for European options.
Let C_n(M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups H_i(C_n(M); Q) are representation stable in the sense of [Church-Farb]. Applying this to the trivial representation, we obtain as a corollary that the un…