The study shows instability of Nikodym maximal function bounds on Riemannian manifolds under metric perturbation.
problem Instability of Nikodym maximal function bounds on Riemannian manifolds under metric perturbation.
method Analyzing the instability of $L^{ rac{d+2}2}$ bounds for the Nikodym maximal function over manifolds of constant sectional curvature and extending to any d-dimensional Riemannian manifold with a local totally geodesic submanifold. result The instability of the bounds for the Nikodym maximal function on Riemannian manifolds under metric perturbation.
Improved bounds for Carleson-Sjölin operators on manifolds with specific curvature conditions.
problem Bounding Carleson-Sjölin operators on manifolds with special curvature conditions.
method Two different methods: one using distance function conditions and the other using contact orders of oscillatory integral operators.
result Improved Lp bounds for Carleson-Sjölin operators on manifolds with constant sectional curvature and those satisfying Sogge's chaotic curvature condition. Study utility maximization with delayed information in continuous time Gaussian markets.
problem Maximizing utility with delayed information in continuous time Gaussian markets.
method Purely probabilistic approach based on Radon-Nikodym derivatives of Gaussian measures.
result Solution for optimal control and value in a specific Gaussian framework.
The paper tackles Kakeya and Nikodym sets on curved manifolds, reducing problems to Euclidean space.
problem Analyzing Kakeya and Nikodym sets on curved manifolds.
method Reduction of problems on curved manifolds to Euclidean space, using Bourgain's condition and recent breakthroughs.
result Establishes the Nikodym conjecture for three-dimensional manifolds with constant sectional curvature.
We estimate Radon-Nikodym derivatives using regularization in reproducing kernel Hilbert spaces.
problem Estimating Radon-Nikodym derivatives in various applications.
method General regularization scheme in reproducing kernel Hilbert spaces.
result High order accuracy in reconstructing Radon-Nikodym derivatives at any point.
A novel approach to interpolation, classification, and clustering using Radon-Nikodym derivatives.
problem Interpolation, classification, and clustering problems in data analysis.
method Radon-Nikodym approach with Lebesgue quadrature for optimal clustering.
result The approach changes both probabilities and the probability space with new observations.
Technical proofs for Radon-Nikodym derivative identities.
problem Formalizing and proving theorems on Radon-Nikodym derivatives.
method Careful consideration of conditional and marginal probability measures.
result New interpretation of mutual and lattum information sums.
We obtain some improved essentially sharp Kakeya-Nikodym estimates for eigenfunctions in two-dimensions. We obtain these by proving stronger related microlocal estimates involving a natural decomposition of phase space that is adapted to the geodesic flow.
The paper optimizes utility for switching models using Lévy processes.
problem Maximizing HARA utilities in Lévy switching models.
method Dual method, f-divergence minimal martingale measures, Hellinger and Kulback-Leibler processes.
result Expressions for optimal strategies and maximal expected utilities.
The execution flow drives market dynamics, validated on real data.
problem Understanding the fundamental driving force of market dynamics.
method Developed a numerical framework using the Radon-Nikodym derivative to calculate execution flow and determined thresholds and characteristic time scales.
result Execution flow is the fundamental driving force of market dynamics.
We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure …
Kernel Density Machines learn probability densities without structural assumptions.
problem Learning probability densities under minimal assumptions.
method Kernel-based framework, agnostic to structural requirements.
result Consistency and functional central limit theorem for sample estimator.
This paper is devoted to the application of B-splines to volatility modeling, specifically the calibration of the leverage function in stochastic local volatility models and the parameterization of an arbitrage-free implied volatility surface calibrated to sparse option data. We use an extension of classical B-splines …
Optimizes sampling from target distributions with applications to online learning.
problem Optimizing the total variation distance between target and sampled distributions.
method Analyzes the sample complexity of approximate rejection sampling and its applications.
result The optimal total variation distance is given by $ ildeΘ( rac{D}{f'(n)})$.
A method for risk valuation using backward stochastic differential equations.
problem Risk evaluation in financial markets.
method Dual representation and stochastic control problem conversion, followed by dynamic programming.
result Piecewise-constant dual control provides a good approximation for risk valuation.
The paper improves sample reweighting methods for adapting to covariate shifts.
problem Improving accuracy in reproducing kernel Hilbert spaces when data distributions differ.
method Combining known error bounds for reweighted kernel regression in RKHS to show reduced sample size needed for accuracy.
result Under weak smoothness conditions, fewer samples are needed for the same accuracy as standard supervised learning.
A new model for estimating multivariate densities efficiently.
problem Estimating complex multivariate densities efficiently.
method CDO model based on kernel mean embeddings and RKHS.
result Competitive performance with neural models and Gaussian processes.
The article provides representations of exchange option prices under SVJD dynamics.
problem Modeling and pricing exchange options under stochastic volatility and jumps.
method Develops representations for European and American exchange options using SVJD dynamics and equivalent martingale measures.
result Derives integro-partial differential equations and representations for exchange option prices.
This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron-Martin type quasi-invariance theorem for the heat kernel measure and give estimates on the Lp norms of the Radon-Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds …
We introduce length dilatation structures on metric spaces, tempered dilatation structures and coherent projections and explore the relations between these objects and the Radon-Nikodym property and Gamma-convergence of length functionals. Then we show that the main properties of sub-riemannian spaces can be obtained f…
For Machine Learning (ML) classification problem, where a vector of x--observations (values of attributes) is mapped to a single y value (class label), a generalized Radon--Nikodym type of solution is proposed. Quantum--mechanics --like probability states ψ2(x) are considered and "Cluster Cente…
We postulates, and then show experimentally, that liquidity deficit is the driving force of the markets. In the first part of the paper a kinematic of liquidity deficit is developed. The calculus-like approach, which is based on Radon--Nikodym derivatives and their generalization, allows us to calculate important chara…
We study heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg-like Lie groups. In particular, we show that Cameron-Martin type quasi-invariance results hold in this subelliptic setting and give Lp-estimates for the Radon-Nikodym derivatives. The main ingredient in our proof is a generalized curvatu…
For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, whil…
The results on the mean-variance hedging problem in Gouriéroux, Laurent and Pham (1998), Rheinländer and Schweizer (1997) and Arai (2005) are extended to discontinuous semimartingale models. When the numéraire method is used, we only assume the Radon-Nikodym derivative of the variance-optimal signed martingale measure …
We provide a necessary and sufficient condition that Lp-norms, 2<p<6, of eigenfunctions of the square root of minus the Laplacian on 2-dimensional compact boundaryless Riemannian manifolds M are small compared to a natural power of the eigenvalue λ. The condition that ensures this is that their L2 norms ove…
Nyström subsampling with Tikhonov regularization for covariate shift adaptation under misspecified case
problem Adaptation to misspecified covariate shift
method Regularized Nyström subsampling with Tikhonov regularization
result Upper bounds on excess risk
A pricing principle is introduced for non-attainable claims in incomplete markets.
problem Pricing non-attainable contingent claims in incomplete markets.
method Distorted Radon-Nikodym derivative and Tsallis relative entropy over a family of equivalent martingale measures.
result The pricing principle is closely related to backward stochastic differential equations and is arbitrage-free and time-consistent.
We study multiple defaults where the global market information is modelled as progressive enlargement of filtrations. We shall provide a general pricing formula by establishing a relationship between the enlarged filtration and the reference default-free filtration in the random measure framework. On each default scena…
Improves bounds on eigenfunctions using microlocal averages in phase space.
problem Improving Lp bounds on eigenfunctions in high frequency limit. method Develops sufficient conditions for microlocal averages in nonpositive curvature and partially hyperbolic flows.
result Improves microlocal averages for eigenfunctions in more general settings.
Develops a new framework for estimating joint probability distributions.
problem Estimating joint probability distributions from large sample sizes.
method Tensor product reproducing kernel Hilbert spaces (RKHS) with normalized and positive model.
result Fast computation and applicability to prediction and classification problems.
New regularization method reduces support of empirical risk minimization solutions.
problem Regularization in empirical risk minimization with relative entropy.
method Introduces Type-II regularization, characterizes solutions, analyzes properties of relative entropy.
result Type-II regularization collapses solution support into reference measure's support.
Quantum groups applied to finance models, extending classical economics.
problem Establishing the relationship between expectation and price in finance.
method Developing quantum group operations and axioms in stochastic and functional calculus.
result Two distinct economic models emerge from the same valuations, extending classical economics.
In a market of deterministic cash flows, given as an additive, symmetric relation of exchangeability on the finite signed Borel measures on the non-negative real time axis, it is shown that the only arbitrage-free price functional that fulfills some additional mild requirements is the integral of the unit zero-coupon b…
We make several improvements to the mean-variance framework for optimal pre-trade algorithmic execution, by working with volume measures and generic price dynamics. Volume measures are the continuum analogies for discrete volume profiles commonly implemented in the execution industry. Execution then becomes an absolute…
Study examines maximal domains of radial harmonic functions across different curvature types.
problem Understanding maximal domains of radial harmonic functions in various curvature settings.
method Analysis of harmonic spaces with positive, zero, and negative curvature.
result Characterization of maximal domains for radial harmonic functions in different curvature contexts.
New theorems show non-embeddability of certain Lie groups and sub-Riemannian manifolds.
problem Non-embeddability of Lie groups and sub-Riemannian manifolds in specific metric spaces.
method Proving non-existence of quasi-isometric embeddings and biLipschitz embeddings into certain metric measure spaces.
result Connected nonabelian nilpotent Lie groups and sub-Riemannian manifolds cannot be embedded into specified metric spaces.
Combines expert models using Kullback-Leibler divergence to create a combined model.
problem Combining expert views on stochastic processes.
method Minimizes weighted Kullback-Leibler divergence to create a barycentre model.
result Existence and uniqueness of the barycentre model with explicit representation.
LEAPS samples discrete distributions via CTMCs and locally equivariant networks.
problem Sampling from discrete distributions with known normalization.
method Continuous-time Markov chain, locally equivariant functions, attention layers, convolutional networks.
result LEAPS minimizes the variance of importance weights, improving sampling efficiency.
Verifies regularity for conditional expectation operators and embeddings, simplifying validation.
problem Characterizing when conditional expectation operators map between function spaces.
method Establishes a verifiable sufficient condition for bounded and Hilbert-Schmidt mappings based on conditional density regularity.
result Averifiable condition for mapping properties of conditional expectation operators simplifies validation.
Differentially private algorithms for submodular maximization under various constraints.
problem Maximizing decomposable submodular functions under constraints while preserving privacy.
method Designing differentially private algorithms for both monotone and non-monotone decomposable submodular maximization under general matroid constraints.
result Improved utility guarantees and competitive performance compared to non-private algorithms.
GONs improve predictions of maximizers from noisy black-box functions.
problem Estimating maximizers of noisy black-box functions.
method Global Optimization Networks (GONs) composed of invertible and unimodal functions.
result GONs outperform convex fits, GPR, and DNNs in prediction accuracy.
New technique improves submodular maximization with barrier functions.
problem Maximizing submodular functions under complex constraints.
method Inspired by barrier functions in continuous optimization, a novel potential function is proposed for approximate minimization.
result Guaranteed 2(k+1+ε)-approximation factor for feasible sets. The paper studies continuous submodular functions and their optimization.
problem Maximizing continuous submodular functions in poly. time.
method Characterization of continuous submodularity, operations preserving it, and algorithms for constrained maximization.
result Continuous submodularity is equivalent to a weak DR property, leading to continuous DR-submodular functions with the full DR property.
The height function of various surfaces decomposes into finite sums of scaled and translated versions of itself.
problem Decomposing the height function of different types of surfaces into simpler components.
method Using Euler-Ramanujan identities and Weierstrass-Enneper representation to decompose height functions of minimal, maximal, timelike minimal, and Born-Infeld surfaces.
result The height function of various surfaces can be expressed as a finite sum of scaled and translated versions of itself.
Fast algorithms developed for adaptive and fully adaptive submodular maximization problems.
problem Maximizing submodular functions subject to constraints in linear time.
method Developed linear-time algorithms for two submodular maximization problems: adaptive and fully adaptive.
result Achieved (1−1/e−ε) approximation ratio for adaptive submodular maximization and $ rac{1-1/e-ε}{4-2/e-2ε}$ for fully adaptive submodular maximization. Differentiable submodular maximization combines learning and optimization.
problem Learning and optimizing submodular functions separately.
method Interpreting greedy maximization as distributions, smoothing, and differentiating.
result The approach optimizes submodular functions with theoretical guarantees.
Study private submodular maximization in streaming data.
problem Private maximization of submodular functions in streaming data.
method Established differentially private baselines and derived better trade-offs for decomposable submodular functions.
result Improved trade-offs between privacy and utility for decomposable submodular functions.