arXiv subjects

Ruodu Wang

Publications and source records attributed to Ruodu Wang.

At least 73 records · Page 4Linked to original sources

arXiv2025

The only admissible way of merging arbitrary e-values

We prove that the only admissible way of merging arbitrary e-values is to use a weighted arithmetic average. This result completes the picture of merging methods for e-values, and generalizes the result of Vovk and Wang (2021, Annals of Statistics, 49(3), 1736--1754) that the only admissible way of symmetrically merging e-values is to use the arithmetic average combined with a constant. Although the proved statement is naturally anticipated, its proof relies on a sophisticated application of optimal transport duality and a minimax theorem.

math.ST↗

arXiv2025

Extremal Dependence Concepts

The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges, and recent developments in extremal dependence concepts have drawn a lot of attention to probability and its applications in several disciplines. The aim of this paper is to review various concepts of extremal positive and negative dependence, including several recently established results, reconstruct their history, link them to probabilistic optimization problems, and provide a list of open questions in this area. While the concept of extremal positive dependence is agreed upon for random vectors of arbitrary dimensions, various notions of extremal negative dependence arise when more than two random variables are involved. We review existing popular concepts of extremal negative dependence given in literature and introduce a novel notion, which in a general sense includes the existing ones as particular cases. Even if much of the literature on dependence is focused on positive dependence, we show that negative dependence plays an equally important role in the solution of many optimization problems. While the most popular tool used nowadays to model dependence is that of a copula function, in this paper we use the equivalent concept of a set of rearrangements. This is not only for historical reasons. Rearrangement functions describe the relationship between random variables in a completely deterministic way, allow a deeper understanding of dependence itself, and have several advantages on the approximation of solutions in a broad class of optimization problems.

stat.ME↗

arXiv2025

The reference interval in higher-order stochastic dominance

Given two random variables taking values in a bounded interval, we study whether one dominates the other in higher-order stochastic dominance depends on the reference interval in the model setting. We obtain two results. First, the stochastic dominance relations get strictly stronger when the reference interval shrinks if and only if the order of stochastic dominance is larger than three. Second, for mean-preserving stochastic dominance relations, the reference interval is irrelevant if and only if the difference between the degree of the stochastic dominance and the number of moments is no larger than three. These results highlight complications arising from using higher-order stochastic dominance in economic applications.

math.PR↗

arXiv2025

Risk Aversion and Insurance Propensity

We provide a new foundation of risk aversion by showing that this attitude is fully captured by the propensity to seize insurance opportunities. Our foundation, which applies to all probabilistically sophisticated preferences, well accords with the commonly held prudential interpretation of risk aversion that dates back to the seminal works of Arrow (1963) and Pratt (1964). In our main results, we first characterize the Arrow-Pratt risk aversion in terms of propensity to full insurance and the stronger notion of risk aversion of Rothschild and Stiglitz (1970) in terms of propensity to partial insurance. We then extend the analysis to comparative risk aversion by showing that the notion of Yaari (1969) corresponds to comparative propensity to full insurance, while the stronger notion of Ross (1981) corresponds to comparative propensity to partial insurance.

econ.TH↗

arXiv2025

Diversification for infinite-mean Pareto models without risk aversion

We study stochastic dominance between portfolios of independent and identically distributed (iid) extremely heavy-tailed (i.e., infinite-mean) Pareto random variables. With the notion of majorization order, we show that a more diversified portfolio of iid extremely heavy-tailed Pareto random variables is larger in the sense of first-order stochastic dominance. This result is further generalized for Pareto random variables caused by triggering events, random variables with tails being Pareto, bounded Pareto random variables, and positively dependent Pareto random variables. These results provide an important implication in investment: Diversification of extremely heavy-tailed Pareto profits uniformly increases investors' profitability, leading to a diversification benefit. Remarkably, different from the finite-mean setting, such a diversification benefit does not depend on the decision maker's risk aversion.

q-fin.PM↗

arXiv2025

The checkerboard copula and dependence concepts

We study the problem of choosing the copula when the marginal distributions of a random vector are not all continuous. Inspired by four motivating examples including simulation from copulas, stress scenarios, co-risk measures, and dependence measures, we propose to use the checkerboard copula, that is, intuitively, the unique copula with a distribution that is as uniform as possible within regions of flexibility. We show that the checkerboard copula has the largest Shannon entropy, which means that it carries the least information among all possible copulas for a given random vector. Furthermore, the checkerboard copula preserves the dependence information of the original random vector, leading to two applications in the context of diversification penalty and impact portfolios. The numerical and empirical results illustrate the benefits of using the checkerboard copula in the calculation of co-risk measures.

q-fin.RM↗

arXiv2025

Cash-subadditive risk measures without quasi-convexity

In the literature on risk measures, cash subadditivity was proposed to replace cash additivity, motivated by the presence of stochastic or ambiguous interest rates and defaultable contingent claims. Cash subadditivity has been traditionally studied together with quasi-convexity, in a way similar to cash additivity with convexity. In this paper, we study cash-subadditive risk measures without quasi-convexity. One of our major results is that a general cash-subadditive risk measure can be represented as the lower envelope of a family of quasi-convex and cash-subadditive risk measures. Representation results of cash-subadditive risk measures with some additional properties are also examined. The notion of quasi-star-shapedness, which is a natural analogue of star-shapedness, is introduced, and we obtain a corresponding representation result via the lower envelope of normalized, quasi-convex and cash-subadditive risk measures.

q-fin.RM↗

arXiv2024

Quantiles under ambiguity and risk sharing

Choquet capacities and integrals are central concepts in decision making under ambiguity or model uncertainty, pioneered by Schmeidler. Motivated by risk optimization problems for quantiles under ambiguity, we study the subclass of Choquet integrals, called Choquet quantiles, which generalizes the usual (probabilistic) quantiles, also known as Value-at-Risk in finance, from probabilities to capacities. Choquet quantiles share many features with probabilistic quantiles, in terms of axiomatic representation, optimization formulas, and risk sharing. We characterize Choquet quantiles via only one axiom, called ordinality. We prove that the inf-convolution of Choquet quantiles is again a Choquet quantile, leading to explicit optimal allocations in risk sharing problems for quantile agents under ambiguity. A new class of risk measures, Choquet Expected Shortfall, is introduced, which enjoys most properties of the coherent risk measure Expected Shortfall. Our theory is complemented by optimization algorithms, numerical examples, and a stylized illustration with financial data.

q-fin.RM↗

arXiv2024

Simultaneous Optimal Transport

We propose a general framework of mass transport between vector-valued measures, which will be called simultaneous optimal transport (SOT). The new framework is motivated by the need to transport resources of different types simultaneously, i.e., in single trips, from specified origins to destinations; similarly, in economic matching, one needs to couple two groups, e.g., buyers and sellers, by equating supplies and demands of different goods at the same time. The mathematical structure of simultaneous transport is very different from the classic setting of optimal transport, leading to many new challenges. The Monge and Kantorovich formulations are contrasted and connected. Existence conditions and duality formulas are established. More interestingly, by connecting SOT to a natural relaxation of martingale optimal transport (MOT), we introduce the MOT-SOT parity, which allows for explicit solutions of SOT in many interesting cases.

econ.TH↗

arXiv2024

Anticomonotonicity for Preference Axioms: The Natural Counterpart to Comonotonicity

Comonotonicity (``same variation'') of random variables minimizes hedging possibilities and has been widely used, e.g., in Gilboa and Schmeidler's ambiguity models. This paper investigates anticomonotonicity (``opposite variation''; abbreviated ``AC''), the natural counterpart to comonotonicity. It minimizes leveraging rather than hedging possibilities. Surprisingly, AC restrictions of several traditional axioms do not give new models. Instead, they strengthen the foundations of existing classical models: (a) linear functionals through Cauchy's equation; (b) Anscombe-Aumann expected utility; (c) as-if-risk-neutral pricing through no-arbitrage; (d) de Finetti's bookmaking foundation of Bayesianism using subjective probabilities; (e) risk aversion in Savage's subjective expected utility. In each case, our generalizations show where the critical tests of classical axioms lie: in the AC cases (maximal hedges). We next present examples where AC restrictions do essentially weaken existing axioms, and do provide new properties and new models.

econ.TH↗

arXiv2024

On the existence of powerful p-values and e-values for composite hypotheses

Given a composite null $ \mathcal P$ and composite alternative $ \mathcal Q$, when and how can we construct a p-value whose distribution is exactly uniform under the null, and stochastically smaller than uniform under the alternative? Similarly, when and how can we construct an e-value whose expectation exactly equals one under the null, but its expected logarithm under the alternative is positive? We answer these basic questions, and other related ones, when $ \mathcal P$ and $ \mathcal Q$ are convex polytopes (in the space of probability measures). We prove that such constructions are possible if and only if $ \mathcal Q$ does not intersect the span of $ \mathcal P$. If the p-value is allowed to be stochastically larger than uniform under $P\in \mathcal P$, and the e-value can have expectation at most one under $P\in \mathcal P$, then it is achievable whenever $ \mathcal P$ and $ \mathcal Q$ are disjoint. More generally, even when $ \mathcal P$ and $ \mathcal Q$ are not polytopes, we characterize the existence of a bounded nontrivial e-variable whose expectation exactly equals one under any $P \in \mathcal P$. The proofs utilize recently developed techniques in simultaneous optimal transport. A key role is played by coarsening the filtration: sometimes, no such p-value or e-value exists in the richest data filtration, but it does exist in some reduced filtration, and our work provides the first general characterization of this phenomenon. We also provide an iterative construction that explicitly constructs such processes, and under certain conditions it finds the one that grows fastest under a specific alternative $Q$. We discuss implications for the construction of composite nonnegative (super)martingales, and end with some conjectures and open problems.

math.ST↗

arXiv2024

Diversification quotient based on expectiles

A diversification quotient (DQ) quantifies diversification in stochastic portfolio models based on a family of risk measures. We study DQ based on expectiles, offering a useful alternative to conventional risk measures such as Value-at-Risk (VaR) and Expected Shortfall (ES). The expectile-based DQ admits simple formulas and has a natural connection to the Omega ratio. Moreover, the expectile-based DQ is not affected by small-sample issues faced by VaR-based or ES-based DQ due to the scarcity of tail data. The expectile-based DQ exhibits pseudo-convexity in portfolio weights, allowing gradient descent algorithms for portfolio selection. We show that the corresponding optimization problem can be efficiently solved using linear programming techniques in real-data applications. Explicit formulas for DQ based on expectiles are also derived for elliptical and multivariate regularly varying distribution models. Our findings enhance the understanding of the DQ's role in financial risk management and highlight its potential to improve portfolio construction strategies.

q-fin.PM↗

arXiv2024

Infinite-mean models in risk management: Discussions and recent advances

In statistical analysis, many classic results require the assumption that models have finite mean or variance, including the most standard versions of the laws of large numbers and the central limit theorems. Such an assumption may not be completely innocent, and it may not be appropriate for datasets with heavy tails (e.g., catastrophic losses), relevant to financial risk management. In this paper, we discuss the importance of infinite-mean models in economics, finance, and related fields, with recent results and examples. We emphasize that many results or intuitions that hold for finite-mean models turn out to fail or even flip for infinite-mean models. Due to the breakdown of standard thinking for infinite-mean models, we argue that if the possibility of using infinite-mean models cannot be excluded, great caution should be taken when applying classic methods that are usually designed for finite-mean cases in finance and insurance.

q-fin.RM↗

arXiv2024

Disappointment concordance and duet expectiles

We introduce an axiom of disappointment-concordance (disco) aversion for a preference relation over acts in an Anscombe-Aumann setting. This axiom means that the decision maker, facing the sum of two acts, dislikes the situation where both acts realize simultaneously as disappointments. Our main result is that, under strict monotonicity and continuity, the axiom of disco aversion characterizes preference relations represented by a new class of functionals belonging to the Gilboa-Schmeidler family, which we call the duet expectiled utilities. When the outcome space is the real line, a duet expectiled utility becomes a duet expectile, which involves two endogenous probability measures. It further becomes a usual expectile, ,a statistical quantity popular in regression and risk measures, when these two probability measures coincide. We discuss properties of duet expectiles and connections with fundamental concepts including probabilistic sophistication, risk aversion, and uncertainty aversion.

econ.TH↗

arXiv2024

Probabilistic risk aversion for generalized rank-dependent functions

Probabilistic risk aversion, defined through quasi-convexity in probabilistic mixtures, is a common useful property in decision analysis. We study a general class of non-monotone mappings, called the generalized rank-dependent functions, which includes the preference models of expected utilities, dual utilities, and rank-dependent utilities as special cases, as well as signed Choquet functions used in risk management. Our results fully characterize probabilistic risk aversion for generalized rank-dependent functions: This property is determined by the distortion function, which is precisely one of the two cases: those that are convex and those that correspond to scaled quantile-spread mixtures. Our result also leads to seven equivalent conditions for quasi-convexity in probabilistic mixtures of dual utilities and signed Choquet functions. As a consequence, although probabilistic risk aversion is quite different from the classic notion of strong risk aversion for generalized rank-dependent functions, these two notions coincide for dual utilities under an additional continuity assumption.

econ.TH↗

arXiv2024

A new characterization of second-order stochastic dominance

We provide a new characterization of second-order stochastic dominance, also known as increasing concave order. The result has an intuitive interpretation that adding a risk with negative expected value in adverse scenarios makes the resulting position generally less desirable for risk-averse agents. A similar characterization is also found for convex order and increasing convex order. The proof techniques for the main result are based on properties of Expected Shortfall, a family of risk measures that is popular in banking and insurance regulation. Applications in risk management and insurance are discussed.

q-fin.RM↗

arXiv2024

Testing mean and variance by e-processes

We address the problem of testing conditional mean and conditional variance for non-stationary data. We build e-values and p-values for four types of non-parametric composite hypotheses with specified mean and variance as well as other conditions on the shape of the data-generating distribution. These shape conditions include symmetry, unimodality, and their combination. Using the obtained e-values and p-values, we construct tests via e-processes, also known as testing by betting, as well as some tests based on combining p-values for comparison. Although we mainly focus on one-sided tests, the two-sided test for the mean is also studied. Simulation and empirical studies are conducted under a few settings, and they illustrate features of the methods based on e-processes.

math.ST↗

arXiv2024

Convolution Bounds on Quantile Aggregation

Quantile aggregation with dependence uncertainty has a long history in probability theory with wide applications in finance, risk management, statistics, and operations research. Using a recent result on inf-convolution of quantile-based risk measures, we establish new analytical bounds for quantile aggregation which we call convolution bounds. Convolution bounds both unify every analytical result available in quantile aggregation and enlighten our understanding of these methods. These bounds are the best available in general. Moreover, convolution bounds are easy to compute, and we show that they are sharp in many relevant cases. They also allow for interpretability on the extremal dependence structure. The results directly lead to bounds on the distribution of the sum of random variables with arbitrary dependence. We discuss relevant applications in risk management and economics.

q-fin.RM↗