Spectral risk measures are attractive risk measures as they allow the user to obtain risk measures that reflect their risk-aversion functions. To date there has been very little guidance on the choice of risk-aversion functions underlying spectral risk measures. This paper addresses this issue by examining two popular …
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48 results for “functional measure”
A new measure quantifies how risk-averse different risk measures are.
problem Measuring the degree of risk aversion among different risk measures.
method Two axioms: normalization and linearity. Two formulas for the functional.
result Quantifies the degree of risk aversion among spectral risk measures.
Study dynamic risk measures and performance indices using distortion functions.
problem Investigate time consistency of dynamic risk measures and performance indices generated by distortion functions.
method Analyze dynamic coherent risk measures (DCRMs) and dynamic weighted value at risk measures, proving their equivalence. Establish properties of families of DCRMs generated by distortion functions and define corresponding dynamic coherent acceptability indices (DCAIs). Examine time consistency of DCRMs and DCAIs.
result DCRM generated by distortion functions are sub-martingale time consistent but not super-martingale time consistent and not weakly acceptance time consistent.
An elementary proof shows submodular functions can be represented as measure suprema.
problem Representing submodular functions as supremum of measures.
method Elementary proof using standard extension theorem of measures.
result Submodular functions can be expressed as supremum of measures.
A framework for sensitivity measures using scoring functions.
problem Constructing sensitivity measures for any elicitable functional.
method Score-based sensitivities constructed via consistent scoring functions.
result Demonstrated intuitive and desirable properties of score-based sensitivities.
Researchers develop a method to infer reference measures from observed functionals.
problem Tackles the challenge of identifying or recovering a reference measure from observed functionals.
method Uses the property of law-invariant functionals defining lower or upper supporting sets in dual spaces of signed measures.
result Illustrates the methodology with examples and develops a modification for Value-at-Risk.
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.
New risk measures for financial and ESG risks using utility functions.
problem Assessing financial and ESG risks using traditional risk measures.
method Developed new risk measures based on utility functions.
result Properties of utility functions translate into properties of risk measures.
Study shows observability from a measurable set for Gevrey functions.
problem Determining observability from a subset for Gevrey functions.
method Used measurable sets and inequalities for Gevrey regular functions.
result Established observability estimates from measurable sets for Gevrey functions.
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.
Paper justifies ideal point forecasts as measurable, clarifying conditions for their existence.
problem Justifying ideal point forecasts as measurable random variables.
method Clarifying and establishing measurability conditions for a wide class of functionals.
result Ideal point forecasts are shown to be measurable, providing theoretical justification.
FFM generates functions between Gaussian and data distributions.
problem Generating functions between Gaussian and data distributions.
method Define a path of measures, learn a vector field to generate this path.
result FFM outperforms other function-space generative models.
This paper shows how to calculate risk measures for sums of two counter-monotonic risks.
problem Calculating risk measures for sums of two counter-monotonic risks.
method Using a fixed distortion function and expressing the risk measure of a sum as the sum of two related measures of the marginals.
result The risk measure of a sum of two counter-monotonic risks can be expressed as the sum of two related distortion risk measures of the marginals.
This paper optimizes functions of probability measures using particle gradient descent for displacement convex functions.
problem Optimizing functions of probability measures with displacement convex properties.
method Particle gradient descent applied to displacement convex functions with theoretical guarantees.
result Finite number of particles and computations are sufficient to find optimal solutions for displacement convex functions.
The paper establishes inequalities and gradient estimates for harmonic functions on Finsler measure spaces.
problem Functional and geometric inequalities on Finsler measure spaces.
method Local uniform Poincaré and Sobolev inequalities, mean value inequality, Harnack inequalities, and gradient estimates.
result Global gradient estimates for positive harmonic functions on Finsler measure spaces.
Identification and scoring functions are statistical tools to assess the calibration and the relative performance of risk measure estimates, e.g., in backtesting. A risk measures is called identifiable (elicitable) it it admits a strict identification function (strictly consistent scoring function). We consider measure…
Proves sufficiency of countable test plans for BV functions on metric spaces.
problem Recovering BV functions and their measures on arbitrary metric spaces.
method Proves sufficiency of countable test plans on arbitrary metric measure spaces and geodesics on spaces.
result Countable test plans are sufficient for BV functions and their measures on metric spaces.
A Similarity Measure Between Functions with Applications to Statistical Learning and Optimizationcs.LG
New measure quantifies function similarity for optimization.
problem Measuring similarity between functions for optimization.
method Quantifies sub-optimality gaps and operation rules.
result Unified measure for various functional similarities.
Paper infers intrinsic dimension from quasi-convex measurements.
problem Inferring intrinsic dimension from measurements by quasi-convex functions.
method Developed a method using filtration of Dowker complexes based on discrete data of point orderings.
result Correct intrinsic dimension can be inferred in the limit of large data under generic assumptions.
Study local minimizers of Ginzburg-Landau functionals in high dimensions, showing energy measures converge to rectifiable measures.
problem Investigating minimizers of Ginzburg-Landau functionals in high dimensions with energy bounds.
method Analyzing minimizers with logarithmic energy bounds and considering the vacuum manifold's homotopy classes.
result Normalized energy measures converge to an -rectifiable measure associated with a stationary varifold.
Simplifies study of multivariate shortfall risk measures.
problem Complexity in studying multivariate shortfall risk measures.
method Defines shortfall risk measures through a 1-dimensional function.
result Simplifies properties of multivariate shortfall risk measures.
Bandit algorithms have been predominantly analyzed in the convex setting with function-value based stationary regret as the performance measure. In this paper, motivated by online reinforcement learning problems, we propose and analyze bandit algorithms for both general and structured nonconvex problems with nonstation…
Spectral risk measures (SRMs) are risk measures that take account of user riskaversion, but to date there has been little guidance on the choice of utility function underlying them. This paper addresses this issue by examining alternative approaches based on exponential and power utility functions. A number of problems…
Dynamic risk measuresq-fin.RM
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 …
Introduces new performance measures using scaled utility functions.
problem Performance measurement in financial contexts.
method Certainty equivalents defined via scaled utility functions, well-posed portfolio optimization problem under generic conditions.
result Link between portfolio dynamics, benchmark process, and utility function choice in the long-run setting.
We investigate the relationship between measurable differentiable structures on doubling metric measure spaces and derivations. We prove: [1] a decomposition theorem for the module of derivations into free modules; [2] the existence of a measurable differentiable structure assuming that one can control the pointwise up…
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.
We study a class of 2-variable polynomials called exact polynomials which contains -polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the …
Researchers calculated EVaR for various distributions using Lambert function.
problem Difficulty in finding analytical representation of EVaR measure.
method Used Lambert function to calculate EVaR for multiple distributions.
result Successfully calculated EVaR for 7 specific distributions.
Paper introduces P-sensitive functions and their applications in robust optimization and financial models.
problem Developing robust models for financial and optimization problems under uncertainty.
method Introducing P-sensitive functions and their localization representations, applying to optimization and financial models.
result P-sensitive functions are precisely those that can be localized, providing a new perspective on robust modeling.
As relational datasets modeled as graphs keep increasing in size and their data-acquisition is permeated by uncertainty, graph-based analysis techniques can become computationally and conceptually challenging. In particular, node centrality measures rely on the assumption that the graph is perfectly known -- a premise …
Let be a complete metric measure space, with a locally doubling measure, that supports a local weak -Poincaré inequality. By assuming a heat semigroup type curvature condition, we prove that Cheeger-harmonic functions are Lipschitz continuous on . Gradient estimates for Cheeger-harmonic func…
Motivated by optimal investment problems in mathematical finance, we consider a variational problem of Neyman-Pearson type for law-invariant robust utility functionals and convex risk measures. Explicit solutions are found for quantile-based coherent risk measures and related utility functionals. Typically, these solut…
Proposes a new dependency function for measuring non-linear relationships.
problem Need for a general-purpose measure of dependency between random variables.
method Revision of ideal properties and proposal of a new dependency function.
result Proposes a new dependency function that meets all desired properties.
New discrepancy function compares discrete probability measures considering space geometry.
problem Comparing discrete probability measures in a geometrically meaningful way.
method Proposes the Fourier Discrepancy Function, proving convexity, differentiability, and providing gradient formula.
result Proves the Fourier Discrepancy is convex, twice differentiable, and provides an explicit gradient formula.
The paper bounds payoffs and option prices in discrete models.
problem Measuring risk in discrete models and incomplete markets.
method Analytical and simulated bounds for payoff functions and option prices.
result Analytical and simulated bounds for European and American options.
Investigates conditional Chisini means and their application to risk measures.
problem Existence of conditional nonlinear means for bounded random variables.
method Defines a mean as a solution to a functional equation induced by T, and provides conditions for the existence of a unique solution.
result Characterizes the scalarization of conditional Risk Measures.
Improves risk and variability measures continuity and consistency.
problem Improving the continuity and consistency of risk and variability measures.
method Analyzes convex and order bounded above functionals on Frechet lattices and Orlicz spaces.
result Order-continuous, law-invariant functionals on Orlicz spaces are strongly consistent everywhere.
Given a measured lamination on a finite area hyperbolic surface we consider a natural measure Mon the real line obtained by taking the push-forward of the volume measure of the unit tangent bundle of the surface under an intersection function associated with the lamination. We show that the measure M gives summation id…
Given a harmonic measure of a hyperbolic lamination on a compact metric space, a positive harmonic function is defined on the universal cover of a typical leaves. We discuss some properties of this function. Especially if all the leaves are hyperbolic, ergodic harmonic measures are divided into two classes.
Submodular risk measuresq-fin.RM
Submodularity is studied for convex risk measures, including Expected Shortfall.
problem Characterizing submodularity in convex risk measures.
method Analyzing submodularity properties of law-invariant coherent risk measures, including Expected Shortfall and Value-at-Risk.
result AES is submodular only when it reduces to ES, and empirical analysis shows AES violations are less frequent than VaR and ES violations.
The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, while the homotopical Dehn function measures fillings of curves by disks. Since the two definitions invo…
Investigates set-valued risk measures for processes and vectors, proving equivalence and providing new dual representations.
problem Investigates set-valued risk measures for processes and vectors.
method Utilizes equivalence of risk measures for processes and vectors and their penalty function formulations.
result Provides new dual representation for risk measures for processes in the set-valued framework.
New proof of energy functional monotonicity via geodesics in measure space.
problem Proving monotonicity of energy functional in generalized Ricci flow.
method Defining adapted cost functional, geodesics, and entropy functional.
result Monotonicity of cost along backwards heat flow and energy functional along generalized Ricci flow.
We study the approximation of measurable functions on the hypercube by functions arising from affine neural networks. Our main achievement is an approximation of any measurable function up to a prescribed precision by a bounded number of neurons, depending only on …
We introduce a new functional measure of tail dependence for weakly dependent (asymptotically independent) random vectors, termed weak tail dependence function. The new measure is defined at the level of copulas and we compute it for several copula families such as the Gaussian copula, copulas of a class of Gaussian mi…
One way to interpret smoothness of a measure in infinite dimensions is quasi-invariance of the measure under a class of transformations. Usually such settings lack a reference measure such as the Lebesgue or Haar measure, and therefore we can not use smoothness of a density with respect to such a measure. We describe h…
Paper finds sparse representation of functions using inverse scale space flow.
problem Finding sparse representation of functions.
method Inverse scale space flow to minimize loss.
result Convergence to optimal solution in ideal and noisy cases.