New approach to model risk measurement using Wasserstein distance.
problem Model risk measurement in financial markets.
method Formulates a new theoretical framework based on Wasserstein distance for non-equivalent probability measures.
result Provides practical results that overcome restrictions of previous methods.
Paper presents a new insurance model equation for diverse structures.
problem Handling diverse insurance models with a single equation.
method Developed a canonical model construction and stochastic backward equations.
result Comparison theorems for different models follow from the new equation.
Study shows Julia sets and gasket limit sets are quasiconformally different.
problem Quasiconformal non-equivalence of Julia sets and gasket limit sets.
method Proved quasiconformal non-equivalence of Julia sets and gasket limit sets.
result Julia sets and gasket limit sets are quasiconformally different.
Study on links formed by pseudocircle arrangements, focusing on three unavoidable cases.
problem Counting non-equivalent positive oriented links with pseudocircle arrangements as shadows.
method Analyzing three unavoidable arrangements of pseudocircles to estimate the number of non-equivalent links.
result Sharp estimates on the number of non-equivalent positive oriented links for the three unavoidable arrangements.
By means of color chord diagrams we establish a necessary and sufficient condition for O-topological equivalence of functions with one essentially critical point on oriented surfaces with edge. We also calculate the number of O-topologically non-equivalent functions with one essentially critical point on oriented s…
This paper develops a dynamic equilibrium model where agents exhibit a strong form of belief heterogeneity: they disagree about zero probability events. It is shown that, somewhat surprisingly, equilibrium exists in this setting, and that the disagreement about nullsets naturally leads to equilibrium asset pricing bubb…
The Gordian distance between two knots measures how many crossing changes are needed to transform one knot into the other. It is known that there are always infinitely many non-equivalent knots `between' a pair of knots of Gordian distance two. In this paper we prove an extreme generalisation of this fact: there are kn…
We consider optimal consumption and portfolio choice in the presence of Knightian uncertainty in continuous-time. We embed the problem into the new framework of stochastic calculus for such settings, dealing in particular with the issue of non-equivalent multiple priors. We solve the problem completely by identifying t…
Bornological metrics on groups are studied, showing equivalence classes and constructing non-equivalent improper metrics.
problem Characterizing and constructing left-invariant metrics on groups that are not proper.
method Introducing bornological metrics and studying their equivalence classes, constructing non-equivalent improper metrics.
result Each coarse equivalence class of bornological metrics is determined by a bornology, and every class contains a canonical left-invariant representative.
Constructs infinitely many non-equivalent wild knots in Menger sponge.
problem Embedding non-equivalent wild knots in the Menger sponge.
method Recursive constructions and geometric approach.
result Controlled set of wild points in a Cantor set within the Menger sponge.
Different space/time splittings lead to non-equivalent norms on spinor bundles.
problem Non-equivalent norms on spinor bundles due to different space/time splittings.
method Exploration of generalized Doppler shift between maximal negative definite subspaces.
result Necessary and sufficient condition for equivalent norms in terms of Doppler shift.
The paper distinguishes non-equivalent branched twist spins using knot determinants.
problem Distinguishing non-equivalent branched twist spins.
method Presented a sufficient condition to distinguish non-equivalent, non-trivial branched twist spins by using knot determinants. Calculated the first elementary ideals and obtained the condition of the knot determinants by substituting -1 for the indeterminate.
result A sufficient condition to distinguish non-equivalent, non-trivial branched twist spins using knot determinants.
Extends risk measure theory to general Orlicz spaces.
problem Applying risk measure theory to non-standard spaces.
method Generalizes results from bounded random variables to general Orlicz spaces, proving new characterizations and extensions.
result Characterizations and extensions of the Fatou property and Kusuoka representation in Orlicz spaces.
Investigates CR structures in 7D, showing 8 is max symmetry dimension.
problem CR structures in 7D with intransitive symmetry.
method Various methods to bound symmetry dimension, demonstrating existence of non-equivalent models.
result Existence of infinitely many non-equivalent submaximally symmetric models.
We prove that for some knot-like objects one can easily recognize non-equivalence w.r.t. all Reidemeister moves by studying some equivalence classes modulo only 2nd Reidemeister moves. There are applications to virtual knots, graph-links and looped graphs.
We explain an algorithm for finding a boundary link Seifert matrix for a given Alexander polynomial. The algorithm depends on several choices and therefore makes it possible to find non-equivalent Seifert matrices for a given Alexander polynomial.
We consider formal deformations of the Poisson algebra of functions (with singularities) on T∗M which are Laurent polynomials of fibers. Tn the case: dimM=1 (M=S1,R), there exists a non-trivial ⋆-product on this algebra non-equivalent to the standard Moyal product.
New quasimorphisms show stable commutator lengths are not equivalent.
problem Equivalence of stable commutator lengths in groups.
method Invariant quasimorphisms for groups acting on the circle.
result Stable commutator lengths are not bi-Lipschitzly equivalent.
Bing-Whitehead Cantor sets were introduced by DeGryse and Osborne in dimension three and greater to produce examples of Cantor sets that were non standard (wild), but still had simply connected complement. In contrast to an earlier example of Kirkor, the construction techniques could be generalized to dimensions bigger…
In this work we consider three problems of the standard market approach to pricing of credit index options: the definition of the index spread is not valid in general, the usually considered payoff leads to a pricing which is not always defined, and the candidate numeraire one would use to define a pricing measure is n…
Study finds six homogeneous surfaces with multiple invariant connections.
problem Characterizing homogeneous surfaces with invariant connections.
method Computed all simply connected homogeneous and infinitesimally homogeneous surfaces.
result Found six non-equivalent surfaces with multiple invariant connections.
When the signed weighted resolution set was defined as an invariant of pseudoknots, it was unknown whether this invariant was complete. Using the Gauss-diagrammatic invariants of pseudoknots introduced by Dorais et al, we show that the signed were-set cannot distinguish all non-equivalent pseudoknots. This goal is achi…
Bayesian approach to robust risk measures under model uncertainty.
problem Representing robust risk measures as a single probability measure.
method Introducing two types of risk measures and analyzing their relation to robust risk measures.
result Robust risk measures can be represented by a mixture probability measure, a Bayesian approach.
Let V be a vector space of dimension n+1. We demonstrate that n-component third-order Hamiltonian operators of differential-geometric type are parametrised by the algebraic variety of elements of rank n in S2(Λ2V) that lie in the kernel of the natural map S2(Λ2V)→Λ4V. Non-equivalent operators corres…
The paper proposes a new method for comparing logistic regression models across different populations.
problem Comparing logistic regression models across sub-populations can lead to misleading results.
method Develops a cascading set of equivalence tests for logistic regression models, addressing coding, predictions, and overall accuracy.
result Equivalence testing incentivizes accurate inference and avoids perverse incentives from significance tests.
We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic branched cover of at most one more knot K' non equivalent to K. To prove the main theor…
The paper explores geometry of probability measures and barycenter maps.
problem Understanding the space of probability measures and their barycenter.
method Information geometry, Fisher metric, dualistic structures, divergences, geodesics.
result Recent developments in the geometry of probability measures and barycenter.
The framework of this paper is that of risk measuring under uncertainty, which is when no reference probability measure is given. To every regular convex risk measure on Cb(Ω), we associate a unique equivalence class of probability measures on Borel sets, characterizing the riskless non positive elements of $…
Extends monetary value measures to probability spaces.
problem No specific problem stated; generalization of monetary value measures.
method Extends category from hi to Prob.
result Generalized monetary value measures to probability spaces.
Found a new compact G2-structure on a 7-manifold.
problem Identifying compact G2-structures on manifolds.
method Lattice in solvable Lie group, Laplacian coflow analysis.
result Obtained a 4-parameter subfamily of expanding solitons.
In this paper we show that any good toric contact manifold has well defined cylindrical contact homology and describe how it can be combinatorially computed from the associated moment cone. As an application we compute the cylindrical contact homology of a particularly nice family of examples that appear in the work of…
Estimating IPM is as hard as estimating under IPM, both requiring similar optimal rates.
problem Estimating Integral Probability Metrics (IPMs) between probability measures.
method Study of minimax optimal rates for IPM estimation and under IPM estimation based on samples.
result Minimax optimal rates for estimating IPM and estimating under IPM are multiplicatively equivalent.
We study a class of nonlinear pricing models which involves the feedback effect from the dynamic hedging strategies on the price of asset introduced by Sircar and Papanicolaou. We are first to study the case of a nonlinear demand function involved in the model. Using a Lie group analysis we investigate the symmetry pro…
The paper reviews historical and modern approaches to asset pricing probability measures.
problem Constructing or selecting probability measures for asset pricing.
method Historical review of various approaches including state price theory, martingale measures, and modern data-driven methods.
result Modern asset pricing involves constructing, transforming, or selecting probability measures to represent market prices.
Regular representation of measures on manifolds with boundary.
problem Representing probability densities on manifolds near boundaries.
method Sufficient conditions for Ck random maps, obstructions to representability. result Conditions for regular representation of measures near boundaries.
A new method quantizes conditional probability measures using deep learning.
problem Quantizing conditional probability measures efficiently.
method DCMQ method using Huber-energy kernel and deep neural network.
result Promising results on various examples.
A scalable approach to learning from probability measures using quantization.
problem Efficiently comparing and manipulating large sets of probability measures.
method Quantization of probability measures to a fixed support, followed by optimal transport computations.
result Consistency and convergence guarantees for quantized measures in various OT-based tasks.
Study shows optimal rates for estimating Wasserstein metric and measures.
problem Minimax optimal estimation of Wasserstein metric between probability measures.
method Analyzes the minimax rates for estimating the Wasserstein-1 metric and probability measures.
result Minimax optimal rates for estimating Wasserstein metric and measures are multiplicatively equivalent.
Study free circle actions on specific 7-manifolds with positive Ricci curvature.
problem Classify and construct metrics on manifolds with a specific cohomology ring.
method Classify manifolds, construct free circle actions, and find positive Ricci curvature metrics.
result Almost all manifolds admit infinitely many non-equivalent free circle actions with positive Ricci curvature.
A new method calculates a barycenter for probability measures using Wasserstein distance.
problem Finding a central measure for a set of probability distributions.
method Regularizing the pushforward measure of a set of probability distributions into the Wasserstein space and then finding the barycenter.
result The method yields a uniquely defined barycenter measure supported on the barycentric points of the input measures.
This paper develops GPCA for probability distributions using Otto-Wasserstein geometry.
problem Analyzing modes of variation in datasets of probability measures.
method Geodesic Principal Component Analysis (GPCA) on Wasserstein space with neural networks.
result Identification of geodesic curves that capture modes of variation in probability distributions.
Investigates the effects of nondominated sets of probability measures in robust models of finance.
problem Uncertainty in financial models due to multiple possible probability measures.
method Analyzes various results from mathematical finance literature under the assumption of nondominated sets of probability measures.
result Many classical results in robust models do not hold when the set of measures is nondominated.
The paper extends Santaló's ellipse measures to hitting probabilities for circle lattices.
problem Calculating hitting probabilities for ellipses intersecting circles.
method Deriving measures for all positions of a moving ellipse inside a fixed circle and calculating hitting probabilities for circle lattices.
result Hitting probabilities for lattices of circles are deduced from ellipse measures.
Classifies positive integral friezes on surfaces.
problem Classifying positive integral friezes on marked bordered surfaces.
method One-to-one correspondence with ideal triangulations and rescaling constants.
result Number of non-equivalent friezes on bordered surfaces is finite.
A new metric for comparing probability measures on graphs, scalable and negative definite.
problem Optimal transport's high complexity and indefiniteness for kernel machines.
method Sobolev transport metric for graph metrics, closed-form formula, negative definiteness.
result Sobolev transport yields a scalable and negative definite metric.
Risk measures are linked to probability structures, and a maximal domain is constructed.
problem Linking risk measures to probability structures and defining a maximal domain.
method Constructing a maximal domain respecting ambiguity and discussing properties.
result A meaningful underlying probability structure is implied by risk measures.
Study convexity properties of gradient map on probability measures.
problem Properties of gradient map on probability measures on submanifolds.
method Analysis of Kähler manifolds and Lie group actions.
result Convexity results for gradient map in Abelian case, extension to non-Abelian case.
Study on identifying probability distributions from random data, showing computable partial learners exist.
problem Identifying probability distributions from random data samples.
method Algorithmic learning theory approach, focusing on computable probability measures and high oracles.
result Characterization of oracles that compute explanatory learners for computable probability measures.