arXiv research

New set-valued star-shaped risk measures introduced for better risk assessment.

Investigates set-valued risk measures for processes and vectors, proving equivalence and providing new dual representations.

Set-valued risk measures on $L^p_d$ with $0 \leq p \leq \infty$ for conical market models are defined, primal and dual representation results are given. The collection of initial endowments which allow to super-hedge a multivariate claim are shown to form the values of a set-valued sublinear (coherent) risk measure. Sc…

Scalar dynamic risk measures for univariate positions in continuous time are commonly represented as backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of set-valued functionals in the recent literature. There are two possible extensi…

Researchers develop multi-utility representations for incomplete preferences linked to risk measures.

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…

Risk measures for multivariate financial positions are studied in a utility-based framework. Under a certain incomplete preference relation, shortfall and divergence risk measures are defined as the optimal values of specific set minimization problems. The dual relationship between these two classes of multivariate ris…

The paper concerns primal and dual representations as well as time consistency of set-valued dynamic risk measures. Set-valued risk measures appear naturally when markets with transaction costs are considered and capital requirements can be made in a basket of currencies or assets. Time consistency of scalar risk measu…

A method for calculating multi-portfolio time consistent multivariate risk measures in discrete time is presented. Market models for $d$ assets with transaction costs or illiquidity and possible trading constraints are considered on a finite probability space. The set of capital requirements at each time and state is c…

The equivalence between multiportfolio time consistency of a dynamic multivariate risk measure and a supermartingale property is proven. Furthermore, the dual variables under which this set-valued supermartingale is a martingale are characterized as the worst-case dual variables in the dual representation of the risk m…

Equivalent characterizations of multiportfolio time consistency are deduced for closed convex and coherent set-valued risk measures on $L^p(Ω,\mathcal F, P; R^d)$ with image space in the power set of $L^p(Ω,\mathcal F_t,P;R^d)$. In the convex case, multiportfolio time consistency is equivalent to a cocycle condition on…

We extend the classical risk minimization model with scalar risk measures to the general case of set-valued risk measures. The problem we obtain is a set-valued optimization model and we propose a goal programming-based approach with satisfaction function to obtain a solution which represents the best compromise betwee…

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…

New risk measures for financial networks avoid external capital, reducing systemic risk.

Revisits superhedging under proportional costs in continuous time markets.

The paper shows vector-valued risk measures ignore dependence structures.

Dual representations for robust risk measures and uncertainty sets.

This work establishes properties on diffeological structures for set-valued maps and measures.

The paper tackles fair set-valued classification under demographic parity constraints.

We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk meas…

New versions of the set-valued average value at risk for multivariate risks are introduced by generalizing the well-known certainty equivalent representation to the set-valued case. The first "regulator" version is independent from any market model whereas the second version, called the market extension, takes trading …

The paper defines and analyzes set-valued stochastic integrals for Lévy processes.

New star-shaped acceptability indexes generalize existing methods.

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…

The time value of money is a critical factor not only in risk analysis, but also in insurance and financial applications. In this paper, we consider a special class of set-valued risk statistics by introducing the time value of money. In fact, the risk statistics established by this method is closer to financial realit…

The paper models and prices cyber insurance risks, distinguishing idiosyncratic, systematic, and systemic risks.

In this paper we present results on dynamic multivariate scalar risk measures, which arise in markets with transaction costs and systemic risk. Dual representations of such risk measures are presented. These are then used to obtain the main results of this paper on time consistency; namely, an equivalent recursive form…

The paper studies the convergence of SAA for systemic risk measures.

We consider a multi-objective risk-averse two-stage stochastic programming problem with a multivariate convex risk measure. We suggest a convex vector optimization formulation with set-valued constraints and propose an extended version of Benson's algorithm to solve this problem. Using Lagrangian duality, we develop sc…

Systemic risk is concerned with the instability of a financial system whose members are interdependent in the sense that the failure of a few institutions may trigger a chain of defaults throughout the system. Recently, several systemic risk measures have been proposed in the literature that are used to determine capit…

One of the crucial problems in mathematical finance is to mitigate the risk of a financial position by setting up hedging positions of eligible financial securities. This leads to focusing on set-valued maps associating to any financial position the set of those eligible payoffs that reduce the risk of the position to …

New approach shows continuity and compactness of martingale measures.

Study online learning with set-valued feedback, showing differences between deterministic and randomized approaches.

Sublinear functionals of random variables are known as sublinear expectations; they are convex homogeneous functionals on infinite-dimensional linear spaces. We extend this concept for set-valued functionals defined on measurable set-valued functions (which form a nonlinear space), equivalently, on random closed sets. …

Choosing a portfolio of risky assets over time that maximizes the expected return at the same time as it minimizes portfolio risk is a classical problem in Mathematical Finance and is referred to as the dynamic Markowitz problem (when the risk is measured by variance) or more generally, the dynamic mean-risk problem. I…

In a capital adequacy framework, risk measures are used to determine the minimal amount of capital that a financial institution has to raise and invest in a portfolio of pre-specified eligible assets in order to pass a given capital adequacy test. From a capital efficiency perspective, it is important to identify the s…

This work establishes uniform convergence of subdifferentials in stochastic optimization.

Introduces epistemic deep learning for better uncertainty estimation in neural networks.

Paper improves conformal prediction for imprecise training data.

Unified framework for set-valued classification tackles ambiguous multi-class datasets.

The paper studies optimal investment using acceptability indices to maximize portfolio performance.

The logcosh loss function helps neural networks learn set-valued functions better.

We propose a novel credit default model that takes into account the impact of macroeconomic information and contagion effect on the defaults of obligors. We use a set-valued Markov chain to model the default process, which is the set of all defaulted obligors in the group. We obtain analytic characterizations for the d…

This paper solves optimal consumption-investment problems with time-varying preferences.

Paper relaxes set-valued prediction in hierarchical classification by considering representation complexity.

Paper proposes set-valued prediction for historical POS tagging.

Generative model for set-valued data using permutation invariant flows.

In most classification tasks there are observations that are ambiguous and therefore difficult to correctly label. Set-valued classifiers output sets of plausible labels rather than a single label, thereby giving a more appropriate and informative treatment to the labeling of ambiguous instances. We introduce a framewo…