Paper explores closedness properties of convex sets in rearrangement invariant spaces.
problem Closedness properties of law-invariant convex sets in rearrangement invariant spaces.
method Analyzes equivalence of different closedness types in rearrangement invariant spaces.
result Order closedness, σ(X,Xn∼)-closedness and σ(X,L∞)-closedness of a law-invariant convex set are equivalent. New principles for collapsing law-invariant functionals to means, extending beyond convexity.
problem Conditions for law-invariant functionals to reduce to means.
method Establishing collapse to the mean principles for non-convex functionals.
result General principles apply beyond convexity, including quasiconvex and Choquet integrals.
New findings on how certain functionals behave in random variable spaces.
problem Understanding when law-invariant convex functionals simplify to the mean.
method Analyzing a broad class of random variable spaces and mild semicontinuity assumptions.
result The expectation functional is the only law-invariant convex functional that collapses to the mean under certain conditions.
The paper characterizes law-invariant star-shaped risk measures.
problem Understanding and characterizing law-invariant star-shaped risk measures.
method Developed characterizations for positively homogeneous and star-shaped functionals, derived Kusuoka-type representations, and offered representations of general law-invariant star-shaped functionals.
result Characterizations of law-invariant star-shaped functionals, including their connections to Value-at-Risk and Expected Shortfall.
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.
We establish general versions of a variety of results for quasiconvex, lower-semicontinuous, and law-invariant functionals. Our results extend well-known results from the literature to a large class of spaces of random variables. We sometimes obtain sharper versions, even for the well-studied case of bounded random var…
Dynamic risk measures follow law invariance principles over time.
problem Tackles dynamic risk measurement principles.
method Shows equivalence between adapted law invariance and recursive one-step conditional-law representation for time-consistent risk measures.
result Identifies adapted law invariance as the dynamic counterpart of ordinary law invariance.
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.
We characterize when a convex risk measure associated to a law-invariant acceptance set in L∞ can be extended to Lp, 1≤p<∞, preserving finiteness and continuity. This problem is strongly connected to the statistical robustness of the corresponding risk measures. Special attention is paid to concre…
We provide a variety of results for (quasi)convex, law-invariant functionals defined on a general Orlicz space, which extend well-known results in the setting of bounded random variables. First, we show that Delbaen's representation of convex functionals with the Fatou property, which fails in a general Orlicz space, c…
Optimal risk sharing without convex preferences using aggregate convexity.
problem Risk sharing among non-convex preferences.
method Aggregate convexity principles and Lyapunov convexity, combined with approximation arguments for law invariant risk measures.
result Derivation of a computationally tractable formula for the conjugate of the value function.
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.
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.
Establishes relationships between prudence and stability properties of risk functionals.
problem Stability properties of risk functionals
method General relationships and preservation of prudence under cash-additive hulls and inf-convolutions
result General methods for constructing prudent risk measures
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.
The paper characterizes risk measures with the Fatou property in function spaces.
problem Investigating the Fatou property of law-invariant risk measures in function spaces.
method Characterization of the Fatou property using the AOCEA property and dual representations.
result Risk measures with the Fatou property exist under the AOCEA property in most classical model spaces.
New concept of partial law invariance connects decision theory and financial risk management.
problem Connecting decision theory and financial risk management under uncertainty.
method Characterizing partially law-invariant coherent risk measures via a novel representation formula.
result Strong partial law invariance bridges the gap between existing risk measure representations.
Introduces Lambda Expected Shortfall as a risk measure generalizing ES.
problem Lack of a comprehensive risk measure that generalizes ES and Lambda-VaR.
method Introduces Lambda-ES, a new risk measure with explicit formula and properties.
result Lambda-ES is the smallest quasi-convex and law-invariant risk measure dominating Lambda-VaR.
A one-to-one correspondence is drawn between law invariant risk measures and divergences, which we define as functionals of pairs of probability measures on arbitrary standard Borel spaces satisfying a few natural properties. Divergences include many classical information divergence measures, such as relative entropy a…
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…
When estimating the risk of a P&L from historical data or Monte Carlo simulation, the robustness of the estimate is important. We argue here that Hampel's classical notion of qualitative robustness is not suitable for risk measurement and we propose and analyze a refined notion of robustness that applies to tail-depend…
This note finds closed-form solutions for mean-risk portfolios using a specific type of mixture distribution.
problem Finding optimal portfolios under mean-risk criteria for general distributions.
method Using normal mean-variance mixture (NMVM) distributions, the paper derives closed-form expressions for mean-risk frontiers by optimizing a Markowitz model with adjusted return vectors.
result Closed-form solutions for mean-risk portfolios are found for return vectors following NMVM distributions.
The regulator is interested in proposing a capital adequacy test by specifying an acceptance set for firms' capital positions at the end of a given period. This set needs to be surplus-invariant, i.e., not to depend on the surplus of firms' shareholders, because the test means to protect firms' liability holders. We pr…
We propose a generalization of the classical notion of the V@Rλ that takes into account not only the probability of the losses, but the balance between such probability and the amount of the loss. This is obtained by defining a new class of law invariant risk measures based on an appropriate family of acceptance set…
Study on efficiency in economies with risk-averse agents, finding Pareto optima.
problem Efficiency in economies with risk-averse agents.
method Analysis of utility functionals, existence and characterization of Pareto optima.
result Existence and comonotone characterization of Pareto optima for risk-averse agents.
Paper analyzes dynamic deviation measures and risk-sharing solutions.
problem Optimal risk-sharing solutions for dynamic deviation measures.
method Dynamic inf-convolution problem involving transformed dynamic deviation measures.
result The only dynamic deviation measure that is law invariant and recursive is variance.
Study examines risk premium convergence rates in risk sharing contracts.
problem Analyzing risk premium convergence rates in risk sharing contracts.
method Examines the limiting behavior of risk premium associated with Pareto optimal risk sharing contracts under general law-invariant risk measures.
result Risk premium convergence rate is typically n1/2, not n. This paper improves the robustness of risk estimation for financial positions.
problem Ensuring robustness of risk measures in the presence of data noise.
method Proposes a quantitative approach using the Fortet-Mourier metric to quantify the variation of true probability measures.
result Derives explicit error bounds for discrepancies between laws of estimators based on true and perturbed data.
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.
Introduces GG-convex risk measures and derives their dual representations.
problem Defining and studying GG-convex risk measures.
method Introduces GG-convex conjugate, derives dual representations, and studies Orlicz risk measures.
result Derives a general dual representation for GG-convex risk measures.
Study finds risk sharing without convexity assumptions.
problem Finding fair risk allocations among agents with heterogeneous beliefs.
method Combines local comonotone improvement with Dieudonné-type argument.
result Existence of Pareto optima without convexity assumption.
Since risky positions in multivariate portfolios can be offset by various choices of capital requirements that depend on the exchange rules and related transaction costs, it is natural to assume that the risk measures of random vectors are set-valued. Furthermore, it is reasonable to include the exchange rules in the a…
Worst-case risk measures refer to the calculation of the largest value for risk measures when only partial information of the underlying distribution is available. For the popular risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), it is now known that their worst-case counterparts can be ev…
Risk measures applied to dynamic Markov processes with varying risk aversion.
problem Investigating dynamic risk measures in Markov decision processes with varying risk aversion.
method Distributional viewpoint on law-invariant convex risk measures, applied to Markov decision processes with latent costs and random actions.
result Existence of optimal policies in finite and infinite time horizons under mild assumptions.
Defines g-expectation of distributions and its applications.
problem Defining g-expectation of distributions. method Two special cases of nonlinear g and law-invariant g-expectation. result Explicit derivation of g-expectation of distributions. Unified framework for risk evaluation under uncertainty.
problem Risk assessment under multiple economic scenarios.
method Axiomatic framework for generalized risk measures.
result Characterization of worst-case, coherent, and robust risk measures.
In this paper, we explore several Fatou-type properties of risk measures. The paper continues to reveal that the strong Fatou property, which was introduced in [17], seems to be most suitable to ensure nice dual representations of risk measures. Our main result asserts that every quasiconvex law-invariant functional on…
Dual representation and properties of expectile-based expected shortfall studied.
problem Studying the expectile-based expected shortfall as a risk measure.
method Provided dual representation in terms of Bochner integral, showed boundedness properties, and computed for selected distributions.
result Explicit dual representation and boundedness properties of expectile-based expected shortfall.
Paper approximates risk measures using SGD with Langevin dynamics.
problem Approximating arbitrary law invariant risk measures.
method Stochastic Gradient Langevin Dynamics (SGD-Langevin) for general risk measures.
result Non-asymptotic convergence rates of the approximation algorithm.
The theory of convex risk functions has now been well established as the basis for identifying the families of risk functions that should be used in risk averse optimization problems. Despite its theoretical appeal, the implementation of a convex risk function remains difficult, as there is little guidance regarding ho…
The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In statistical decision theory, risk measures for which such verification and comparison is possible,…
New risk measure extensions preserve key properties.
problem Extending risk measures to larger spaces while preserving properties.
method Unique extension of dilatation monotone risk measures to L1. result Risk measures extend uniquely and preserve monotonicity, convexity, and cash-additivity.
Study provides convergence rates for risk measure estimation.
problem Estimating risk measures from limited data.
method Plug-in estimation using empirical measures.
result Non-asymptotic convergence rates for risk measure estimation.
This work generalizes Hamiltonian mechanics using closed differential forms.
problem Hidden invariants in classical Hamiltonian mechanics.
method Establishes a novel correspondence between generalized Hamiltonian mechanics and multisymplectic geometry.
result Key theorems linking classical and generalized Hamiltonian systems.
Expanding on techniques of concentration of measure, we develop a quantitative framework for modeling liquidity risk using convex risk measures. The fundamental objects of study are curves of the form (ρ(λX))λ≥0, where ρ is a convex risk measure and X a random variable, and we call such a curve a \emph{liqu…
We consider families of strongly consistent multivariate conditional risk measures. We show that under strong consistency these families admit a decomposition into a conditional aggregation function and a univariate conditional risk measure as introduced Hoffmann et al. (2016). Further, in analogy to the univariate cas…
Assuming that agents' preferences satisfy first-order stochastic dominance, we show how the Expected Utility paradigm can rationalize all optimal investment choices: the optimal investment strategy in any behavioral law-invariant (state-independent) setting corresponds to the optimum for an expected utility maximizer w…
A new AMM design reduces impermanent loss and retains more liquidity.
problem Inefficiencies in conventional AMM designs lead to liquidity loss and user engagement issues in DEXs.
method Proposes a dual-mechanism framework: a power-law invariant BMM and dynamic rebate system.
result Reduces impermanent loss by 36% and retains 3.98x more liquidity during price volatility.