We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.
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48 results for “convex valuations”
Researchers classify and decompose valuations on convex functions.
problem Classifying valuations on convex functions.
method Geometric decomposition of valuations, using properties of special subspaces and Monge-Ampère-type operators.
result Valuations decompose into subspaces defined by vanishing properties.
We study convex risk measures describing the upper and lower bounds of a good deal bound, which is a subinterval of a no-arbitrage pricing bound. We call such a convex risk measure a good deal valuation and give a set of equivalent conditions for its existence in terms of market. A good deal valuation is characterized …
In this paper, we endow the space of continuous translation invariant valuation on convex sets generated by mixed volumes coupled with a suitable Radon measure on tuples of convex bodies with two appropriate norms. This enables us to construct a continuous extension of the convolution operator on smooth valuations to n…
Proves hard Lefschetz theorem and Hodge-Riemann relations for convex valuations.
problem Proving properties of convex valuations analogous to Kähler manifolds.
method Elliptic operator theory and perturbation theory applied to unbounded operators on a Hilbert space.
result Establishes hard Lefschetz theorem and Hodge-Riemann relations for convex bodies.
New space for valuations in non-Archimedean setting with duality properties.
problem Developing a non-Archimedean analogue of classical valuation spaces.
method Construction of a new space of valuations with similar structures to classical spaces.
result The new space satisfies Poincaré duality and hard Lefschetz theorem.
New proof of Alesker's Irreducibility Theorem using localization techniques.
problem Representing polynomial valuations on convex bodies.
method Introducing a localization technique for polynomial valuations and reducing to a representation problem for differential forms.
result Smooth and translation invariant valuations are representable by integration with the normal cycle.
We study the properties of the multiplicative structure on valuations on convex sets. We prove a new version of the hard Lefschetz theorem for even translation invariant continuous valuations, and discuss related problems of integral geometry. Then we formulate a conjectural analogue of this result for odd valuations.
Existence of smooth valuations on subspaces is shown for certain conditions.
problem Existence of smooth valuations on subspaces with given restrictions.
method Analyzing compatibility and using recursive descriptions of the cosine transform.
result Compatibility is sufficient for extensibility in certain regimes.
-Steiner quermassintegralsmath.DG
New -Steiner quermassintegrals defined from Steiner formula.
problem Defining new -Steiner quermassintegrals.
method Analogy to classical Steiner formula, investigating properties in convex bodies.
result Rotation and reflection invariant valuations in convex bodies.
The decomposition of the space of continuous and translation invariant valuations into a sum of SO(n) irreducible subspaces is obtained. A reformulation of this result in terms of a Hadwiger type theorem for continuous translation invariant and SO(n)-equivariant tensor valuations is also given. As an application, symme…
Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuationsmath.MG
We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations. It implies new…
Proves Hodge-Riemann relations for mixed valuations and strengthens geometric inequalities.
problem Geometric inequalities and mixed Hodge-Riemann relations for translation-invariant valuations.
method Proves mixed Hodge-Riemann relations for various convex bodies and their mixed volumes.
result Strengthened geometric inequalities for lower dimensional convex bodies.
This paper approaches the definition and properties of dynamic convex risk measures through the notion of a family of concave valuation operators satisfying certain simple and credible axioms. Exploring these in the simplest context of a finite time set and finite sample space, we find natural risk-transfer and time-co…
We show that the natural "convolution" on the space of smooth, even, translation-invariant convex valuations on a euclidean space , obtained by intertwining the product and the duality transform of S. Alesker, may be expressed in terms of Minkowski sum. Furthermore the resulting product extends naturally to odd valu…
An overview of some of the recent developments in the theory of valuations on convex sets and its generalizations to manifolds is given. The exposition is focused towards applications to integral geometry; several of such applications are discussed.
The spaces of Sp(n)-, Sp(n)U(1)- and Sp(n)Sp(1)- invariant, translation invariant, continuous convex valuations on the quaternionic vector space H^n are studied. Combinatorial dimension formulas involving Young diagrams and Schur polynomials are proved.
Paper refines Alesker-Bernig-Schuster theorem, proving Hodge-Riemann relations for Euclidean balls.
problem Understanding translation-invariant valuations and their geometric implications.
method Explicit construction of highest weight vectors and analysis of natural operations on these vectors.
result Proof of Hodge-Riemann relations for Euclidean balls, extending geometric inequalities.
New loss functions optimize pricing policies using transaction data, ensuring expected revenue guarantees.
problem Optimizing pricing policies with transaction data where valuation data is not directly observed.
method Introducing convex loss functions for contextual pricing, focusing on log-concave valuation distributions.
result Proved expected revenue bounds for generalized hinge and quantile pricing loss functions.
This article presents valuation of Treasury Bonds (T-Bonds) on Macedonian Stock Exchange (MSE) and empirical test of duration, modified duration and convexity of the T-bonds at MSE in order to determine sensitivity of bonds prices on interest rate changes. The main goal of this study is to determine how standard valuat…
The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area measures are shown to form a module over smooth valuations and the module of unitarily …
The projection body operator Π, which associates with every convex body in Euclidean space Rn its projection body, is a continuous valuation, it is invariant under translations and equivariant under rotations. It is also well known that Π maps the set of polytopes in Rn into itself. We show that Π is the only non-trivi…
Hadwiger's Theorem states that Euclidean-invariant convex-continuous valuations of definable sets are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable real-valued functions on n-dimensional Euclidean space. This generalizes intrinsic vol…
Hermitian integral geometrymath.DG
We give in explicit form the principal kinematic formula for the action of the affine unitary group on $\C^n$, together with a straightforward algebraic method for computing the full array of unitary kinematic formulas, expressed in terms of certain convex valuations introduced, essentially, by H. Tasaki. We introduce …
This paper studies convex duality in optimal investment and contingent claim valuation in markets where traded assets may be subject to nonlinear trading costs and portfolio constraints. Under fairly general conditions, the dual expressions decompose into tree terms, corresponding to the agent's risk preferences, tradi…
Fixed points of mean section operators found in convex bodies.
problem Characterizing fixed points of mean section operators in convex bodies.
method Characterization of rotation equivariant operators using spherical Laplacian mass distribution, and application of Minkowski valuations.
result Euclidean balls are the only fixed points of mean section operators in a neighborhood of the unit ball.
We investigate the structure of good deal bounds, which are subintervals of a no-arbitrage pricing bound, for financial market models with convex constraints as an extension of Arai and Fukasawa (2014). The upper and lower bounds of a good deal bound are naturally described by a convex risk measure. We call such a risk…
Researchers explore valuations on polyhedra and topological arrangements without imposing algebraic structures.
problem Understanding valuations on polyhedra and their connections to topological arrangements.
method Generalizes the setting of valuations on convex polyhedra to collections of defining hyperplanes without imposing algebraic structures.
result Uncovered a close relationship between scissors congruence problems and finite hyperplane arrangements.
New proof confirms operations on constructible functions match theory.
problem Matching operations on constructible functions with generalized valuations theory.
method Comparison with characteristic cycles approach.
result Operations on constructible functions match generalized valuations theory under mild assumptions.
Study on affine surface areas and their inequalities for convex bodies.
problem Understanding weighted affine surface areas in convex bodies.
method Investigating valuations, isoperimetric inequalities, and connections to divergences.
result Established isoperimetric inequalities for weighted affine surface areas.
Valuations constitute a class of functionals on convex bodies which include the Euler-characteristic, the surface area, the Lebesgue-measure, and many more classical functionals. Curvature measures may be regarded as "localised`` versions of valuations which yield local information about the geometry of a body's bounda…
New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.
problem Defining and analyzing surface area measures for ball-convex bodies.
method Introducing relative surface areas, proving invariance and inequalities, and using geometric interpretations.
result Established inequalities and a new notion of entropy for ball-convex bodies.
We study robust notions of good-deal hedging and valuation under combined uncertainty about the drifts and volatilities of asset prices. Good-deal bounds are determined by a subset of risk-neutral pricing measures such that not only opportunities for arbitrage are excluded but also deals that are too good, by restricti…
Probabilistic theory counts intersections in Riemannian spaces.
problem Counting intersections in Riemannian homogeneous spaces.
method Introduces probabilistic intersection ring , a graded commutative and associative real Banach algebra.
result Probabilistic intersection ring structure defined for spheres, real projective space, and complex projective space.
A celebrated theorem of Hadwiger states that the Euler-Poincaré characteristic is the the unique invariant and continuous valuation on the distributive lattice of compact polyhedra in R^n that assigns value one to each convex non-empty such polyhedron. This paper provides an analogue of Hadwiger's result for finitely p…
This memoir presents a systematic study of the utility maximization problem of an investor in a constrained and unbounded financial market. Building upon the work of Hu et al. (2005) [Ann. Appl. Probab., 15, 1691--1712] in a bounded framework, we extend our analysis to the more challenging unbounded case. Our methodolo…
Defining, Estimating and Using Credit Term Structures. Part 1: Consistent Valuation Measuresq-fin.PR
In this three-part series of papers, we argue that the conventional spread measures are not well defined for credit-risky bonds and introduce a set of credit term structures which correct for the biases associated with the strippable cash flow valuation assumption. We demonstrate that the resulting estimates are signif…
S. Alesker has shown that if is a compact subgroup of O(n) acting transitively on the unit sphere then the vector space of continuous, translation-invariant, -invariant convex valuations on has the structure of a finite dimensional graded algebra over satisfying Poincare duality. We s…
Actuarial-consistency and two-step actuarial valuations: a new paradigm to insurance valuationq-fin.PR
Paper introduces new actuarial-consistent valuations for insurance liabilities.
problem Valuation of insurance liabilities considering both financial and actuarial risks.
method Proposes two-step actuarial valuations and actuarial-consistent procedures.
result Actuarial-consistent valuations are equivalent to two-step actuarial valuations under coherence.
In the second part of our series we suggest new definitions of credit bond duration and convexity that remain consistent across all levels of credit quality including deeply distressed bonds and introduce additional risk measures that are consistent with the survival-based valuation framework. We then show how to use t…
Valuation Reveals Uncertaintyq-fin.MF
Paper recovers uncertainty from dynamic valuation rules.
problem Recovering latent uncertainty from observable valuation rules.
method Developed procedures to identify and characterize uncertainty structures from valuation rules.
result Valuation rules contain sufficient information to identify and recover uncertainty structures.
This paper examines the valuation of a generalized American-style option known as a Game-style call option in an infinite time horizon setting. The specifications of this contract allow the writer to terminate the call option at any point in time for a fixed penalty amount paid directly to the holder. Valuation of a pe…
Study convolution of invariant valuations on Lie groups.
problem Understanding convolution of valuations on Lie groups.
method Explicit formula for left-invariant valuations, showing existence of smooth bi-invariant valuations, defining convolution on arbitrary Lie groups.
result Unified convolution operations on Lie groups.
SL(n) covariant valuations on Orlicz spaces are represented and characterized.
problem Representing SL(n) covariant valuations on Orlicz spaces.
method Representation theorem established for continuous, SL(n) covariant vector-valued valuations.
result Unique characterization of SL(n) covariant valuations as moment vectors.
We propose a projected gradient dynamical system as a model for a bargaining scheme for an asset for which the two interested agents have personal valuations which do not initially coincide. The personal valuations are formed using subjective beliefs concerning the future states of the world and the reservation prices …
Market valuation duration is 175 years, but drops to 46 years during crises.
problem Understanding the duration of market valuation and its impact on returns.
method Comparing market valuation ratios and dividends to estimate duration, analyzing the discount rate effect.
result Valuation duration is negatively correlated with market returns, with a robust out-of-sample R2 of 15%.
Paper simplifies default process modeling and credit valuation.
problem Modeling and pricing derivative securities with credit risk.
method Integrates default process, probability, and correlation into a unified framework.
result Risky valuation is Martingale in the proposed model.
Business-cycles and Cash-on-Market: Pre-money Startup Valuation in the Macroeconomic Environmentq-fin.GN
Business cycles affect startup valuations, both directly and indirectly.
problem How do business cycles impact startup valuations?
method Structural Equation Model approach using a dataset of 1,089 venture capital investments.
result Business cycles impact startup valuations both directly and indirectly.