Unified framework for robust risk measures beyond convexity.
problem Developing risk measures for uncertainty beyond classical convexity.
method Constructing robust quasi-convex measures through uncertainty sets.
result Unified framework for robust quasi-convex risk measures.
The paper studies martingales and super-martingales under a convex set of measures.
problem Understanding martingales and super-martingales in a convex set of equivalent measures.
method Introduced local regular super-martingales and proved necessary and sufficient conditions for their regularity.
result Generalized Doob's decomposition theorem for super-martingales under a convex set of measures.
New set-valued star-shaped risk measures introduced for better risk assessment.
problem Improving risk assessment in financial contexts.
method Developed new set-valued star-shaped risk measures and proved their representation theorems.
result Set-valued star-shaped risk measures can be represented as unions of set-valued convex risk measures.
Paper introduces risk measures for non-convex portfolios.
problem Risk measurement in non-convex transaction costs models.
method Analyzes all portfolio selections to find acceptable positions.
result Properties and examples of non-convex portfolio risk measures.
The paper refines and generalizes worst-case law invariant convex risk measures.
problem Developing robust convex risk measures under uncertainty sets.
method Generalizing closed forms for worst-case law invariant convex risk measures with uncertainty sets based on norms and moment constraints.
result Explicit closed forms for convex risk measures are developed and assessed through numerical simulations.
Paper introduces a continuous convexity measure for compact sets.
problem Lack of continuity in existing convexity measures.
method Enriched axioms with continuity hypothesis in Hausdorff's sense.
result Theoretical grounding and continuous convexity measure construction.
Paper introduces quasi-logconvex risk measures and their properties.
problem Characterizing and understanding new risk measures.
method Characterization through dual representation and properties of acceptance sets.
result Established dual representation and taxonomy of quasi-logconvex risk measures.
Dual representations for robust risk measures and uncertainty sets.
problem Characterizing continuity of robust risk measures and their uncertainty sets.
method Develop dual representations for robust risk measures and uncertainty sets based on distinct geometric assumptions.
result Two dual frameworks for consolidated uncertainty sets are complementary, not interchangeable.
Study geodesic distances and convexity in contact sets.
problem Understanding geodesic distances and convexity in contact sets.
method Extending results on quasi-psh functions and big cohomology classes, studying Monge-Ampère measures on contact sets.
result Convexity of the K-energy in big and nef cohomology classes.
Constructs new elicitable risk measures with multiplicative scoring functions.
problem Defining new elicitable risk measures with specific properties.
method Constructs new elicitable risk measures using a multiplicative scoring function.
result Encompasses and allows construction of novel elicitable risk measures.
We introduce a particular class of unbounded closed convex sets of Rd+1, called F-convex sets (F stands for future). To define them, we use the Minkowski bilinear form of signature (+,...,+,−) instead of the usual scalar product, and we ask the Gauss map to be a surjection onto the hyperbolic space $\H^d$. Impo…
We prove some results concerning the boundary of a convex set in $\H^n$. This includes the convergence of curvature measures under Hausdorff convergence of the sets, the study of normal points, and, for convex surfaces, a generalized Gauss equation and some natural characterizations of the regular part of the Gaussian …
Introduces Star-Shaped deviation measures for risk analysis.
problem Risk measurement and analysis in finance.
method Characterizes Star-Shaped deviation measures through acceptance sets and convex deviation measures.
result Exposes the relationship between Star-Shaped risk measures and deviation measures.
In this work we study the Lebesgue property for convex risk measures on the space of bounded càdlàg random processes (R∞). Lebesgue property has been defined for one period convex risk measures in \cite{Jo} and earlier had been studied in \cite{De} for coherent risk measures. We introduce and study th…
This paper gives an overview of the theory of dynamic convex risk measures for random variables in discrete time setting. We summarize robust representation results of conditional convex risk measures, and we characterize various time consistency properties of dynamic risk measures in terms of acceptance sets, penalty …
Simple conditions for comonotonic additive risk measures from acceptance sets.
problem Conditions for comonotonic additive risk measures from acceptance sets.
method Conditions on acceptance sets for induced comonotonic additive risk measures.
result Acceptance sets induce comonotonic additive risk measures if and only if the acceptance sets and their complements are stable under convex combinations of comonotonic random variables.
The paper proves a Gaussian measure's concavity for symmetric convex sets up to a factor of 2.
problem Proving the concavity of the Gaussian measure for symmetric convex sets.
method Analyzing the conjecture of Gardner and Zvavitch, proving the inequality up to a factor of 2.
result The Gaussian measure satisfies a factor of 2 concavity inequality for symmetric convex sets.
The paper characterizes dynamic return and star-shaped risk measures via BSDEs.
problem Characterizing dynamic return and star-shaped risk measures.
method Characterization of star-shaped functionals and BSDEs.
result Existence of convex BSDEs with non-empty set of supersolutions.
In the present contribution we characterize law determined convex risk measures that have convex level sets at the level of distributions. By relaxing the assumptions in Weber (2006), we show that these risk measures can be identified with a class of generalized shortfall risk measures. As a direct consequence, we are …
Set risk measures extend traditional risk measures to handle sets of positions.
problem Handling sets of positions with a single capital requirement.
method Developed an axiomatic framework for set risk measures, dual representation through topology and measures.
result Characterized worst-case set risk measures and provided examples.
In the paper, we introduce the notion of a local regular supermartingale relative to a convex set of equivalent measures and prove for it the necessary and sufficient conditions of optional Doob decomposition in the discrete case. This Theorem is a generalization of the famous Doob decomposition onto the case of superm…
New curvature measures characterize non-convex Wulff shapes in normed spaces.
problem Characterizing non-convex sets with curvature measures.
method Extending curvature measures to non-convex and non-smooth sets in normed spaces.
result Finite unions of disjoint Wulff shapes are the only sets with proportional curvature measures.
The paper solves a problem of prescribing curvature measures on convex domains.
problem Given a measure, find a convex domain with a specific curvature measure.
method Analyzes the solvability and uniqueness of convex domains for prescribed curvature measures.
result The problem is solvable if and only if the measure has a specific property, and the solution is unique up to translation.
Unique entropy measure found for convex projective manifolds.
problem Entropy measure for convex projective manifolds.
method Developed Patterson--Sullivan densities and mixing theory.
result Unique mixing measure of maximal entropy exists.
Solves Alexandrov's problem for hyperbolic convex bodies.
problem Finding a convex body with a given curvature measure in hyperbolic space.
method Defined Gauss curvature measure, proved existence and uniqueness of solution.
result Uniqueness of the solution to Alexandrov's problem in hyperbolic space.
New risk measures improve machine learning robustness.
problem Improper estimation of data distribution leads to poor out-of-sample performance in machine learning.
method Developed a new framework from quantitative finance to recast the min-max problem as a convex minimization problem.
result Proposed efficient algorithms to solve the convex optimization problems involving complex constraints.
To provide a solid analytic foundation for the module approach to conditional risk measures, this paper establishes a complete random convex analysis over random locally convex modules by simultaneously considering the two kinds of topologies (namely the (ε,λ)--topology and the locally L0-- convex topolo…
The paper studies market viability and completeness in discrete markets.
problem Characterizing the set of equivalent martingale measures in finite markets.
method Characterization as convex combinations of martingale measures, algorithm for finding these measures.
result Limitations of using discrete-time models to understand continuous-time models.
Proposes new deviation measures using Minkowski gauges.
problem Lack of suitable acceptance sets for deviation measures.
method Derives deviation measures through Minkowski gauges of acceptable sets.
result Any positive homogeneous deviation measure can be accommodated in the framework.
In the paper, we introduce the notion of a local regular supermartingale relative to a convex set of equivalent measures and prove for it an optional Doob decomposition in the discrete case. This Theorem is a generalization of the famous Doob decomposition onto the case of supermartingales relative to a convex set of e…
The paper extends localisation technique to multiple constraints in Euclidean spaces.
problem Proving log-concavity of conditional measures in decomposed convex sets.
method Defining partitions of maximal closed convex sets and proving log-concavity of conditional measures.
result Existence of a partition and log-concavity of conditional measures for almost every set of the partition.
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 …
Solves Christoffel problem for disk area measures on spheres.
problem Conditions for a measure to be a disk area measure of convex bodies.
method Integral representation and differential equation reformulation.
result Reconstructs support function from disk area measure.
Equivalent characterizations of multiportfolio time consistency are deduced for closed convex and coherent set-valued risk measures on Lp(Ω,F,P;Rd) with image space in the power set of Lp(Ω,Ft,P;Rd). In the convex case, multiportfolio time consistency is equivalent to a cocycle condition on…
Sharp upper bounds on inscribed radius for metric spaces with convex boundary.
problem Bounding inscribed radius in metric measure spaces with convex boundary.
method Proves sharp upper bounds on inscribed radius for subsets with convex boundary.
result Sharp upper bounds on inscribed radius for subsets with convex boundary.
Extends inf-convolution to countable risk measures for risk sharing.
problem Limited inf-convolution theory to finite sets of risk measures.
method Extends inf-convolution to countable sets, investigates properties and results.
result Generalizes known properties and results to countable case.
Paper discusses natural quasiconvexity and its relation to decomposable sums in risk measures.
problem Understanding natural quasiconvexity and its implications in risk measures.
method Relates natural quasiconvexity to decomposable sums, proposes a general treatment of convexity index, and proves equivalence for certain spaces.
result Natural quasiconvexity and convexity are equivalent for conditional risk measures on Lp spaces under mild conditions. For convex co-compact hyperbolic manifolds Γ\Hn+1 for which the dimension of the limit set satisfies δΓ<n/2, we show that the high-frequency Eisenstein series associated to a point ξ "at infinity" concentrate microlocally on a measure supported by (the closure of) the set of points in the …
The risk of financial positions is measured by the minimum amount of capital to raise and invest in eligible portfolios of traded assets in order to meet a prescribed acceptability constraint. We investigate nondegeneracy, finiteness and continuity properties of these risk measures with respect to multiple eligible ass…
Extends dual volume and curvature measures to broader functions and sets, solving Minkowski problems.
problem Characterize measures for which there exists a convex body with a given dual Orlicz curvature measure.
method Extends dual volume and curvature measures to broader functions and sets, proving existence and existence of solutions for Minkowski problems.
result Existence of convex polytopes and solutions for Minkowski problems when measures are discrete or even.
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 $…
The paper proves a rigidity theorem for non-compact convex sets in hyperbolic 3-space.
problem Determining a closed convex set in hyperbolic 3-space by its boundary metric.
method Pogorelov's rigidity theorem, Hausdorff measure, and complex analysis techniques.
result The intrinsic path metric on the boundary determines a closed convex set up to isometry under certain conditions.
Paper characterizes monotonic mean-deviation risk measures.
problem Developing consistent risk measures from mean-deviation models.
method Applying a risk-weighting function to the deviation part of a mean-deviation model.
result Characterizes monotonic mean-deviation measures as consistent risk measures.
The paper analyzes elicitability of return risk measures and their scoring functions.
problem Elicitability of return risk measures and their scoring functions.
method Dual representation results for convex and geometrically convex return risk measures, axiomatic characterizations of Orlicz premia, and construction of strictly consistent scoring functions.
result Orlicz premia are the only elicitable return risk measures under different sets of conditions.
The paper shows vector-valued risk measures ignore dependence structures.
problem Defining capital allocation rules for random vectors with dependence.
method Defined vector-valued risk measures by axioms and showed their properties.
result Vector-valued risk measures ignore dependence structures, unlike set-valued measures.
We provide a characterization in terms of Fatou closedness for weakly closed monotone convex sets in the space of P-quasisure bounded random variables, where P is a (possibly non-dominated) class of probability measures. Applications of our results lie within robust versions the Fundamental Theo…
We study compressing empirical measures in finite RKHSs using convex optimization.
problem Efficiently approximating empirical measures in high-dimensional spaces.
method Convex optimization and lower bounds on ball size.
result High probability lower bounds on ball size under various conditions.
The paper explores non-convex risk measures and their characterizations.
problem Characterizing non-convex risk measures without convexity or weak convexity.
method Characterizes monetary risk measures as lower envelopes of families of convex or coherent risk measures, considering law-invariance and SSD-consistency.
result Unified representation theorems for law-invariant risk measures, including VaR.