Convex Order Research Articles

Backward and forward Wasserstein projections in stochastic order

Motivated by applications of stochastic orders in statistics and economics, we study metric projections onto cones in the Wasserstein space of probability measures, defined by stochastic orders. Duality theorems for backward and forward projections are established under mild conditions, which lay a foundation for investigating sampling of measures in stochastic orders. To demonstrate applications of these theorems beyond practice, particular attention is given to mathematical properties of convex order and subharmonic order. While backward and forward cones possess distinct geometric properties, strong connections between backward and forward projections can be obtained in the convex order case. Compared with convex order, the study of subharmonic order is subtler, particularly the existence and regularity of optimal mappings. In all cases, Brenier-Strassen type polar factorization theorems are proved, thus providing a full picture of the decomposition of optimal couplings between probability measures given by deterministic contractions (resp. expansions) and stochastic couplings. The factorization of convex order projection completes the decomposition obtained by Gozlan and Juillet, which builds a connection with Caffarelli's contraction theorem. A further noteworthy addition to earlier results is the decomposition in the subharmonic order case where the optimal mappings are characterized by volume distortion properties. To our knowledge, this is the first time in this occasion such results are available in the literature.

The NBRULC Reliability Class: Mathematical Theory and Goodness-of-Fit Testing with Applications to Asymmetric Censored and Uncensored Data

The majority of approaches proposed in the past few decades to solve life test problems have differed markedly from those used for closely related, yet broader, issues. Due to the complexity of data that are generated each day in many practical domains, as a result of the development of scales for rating the success or failure of reliability, a new domain of reliability has been created. This domain is referred to as life classes, where specific probability distributions are presented. In this study, it is shown that the use of the quality-of-fit technique to solve problems involving life testing makes sense, and produces simpler processes that are roughly equivalent or superior to those used in traditional procedures. They may also behave better in limited samples. This work investigates a novel quality-of-fit test statistic; it is based on an exponential transform and is compared to the best renewal used Laplace test in increasing convex ordering (NBRULC). Evidence for approach normality is provided. The calculated variables include powers, Pitman asymptotic effectiveness, and critical points. Methods on how to handle censored data were also studied. Our experiments have real-world applications in the fields of medicine and engineering.

Applications of q-Calculus Multiplier Operators and Subordination for the Study of Particular Analytic Function Subclasses

In this article, a new linear extended multiplier operator is defined utilizing the q-Choi–Saigo–Srivastava operator and the q-derivative. Two generalized subclasses of q—uniformly convex and starlike functions of order δ—are defined and studied using this new operator. Necessary conditions are derived for functions to belong in each of the two subclasses, and subordination theorems involving the Hadamard product of such particular functions are stated and proven. As applications of those findings using specific values for the parameters of the new subclasses, associated corollaries are provided. Additionally, examples are created to demonstrate the conclusions’ applicability in relation to the functions from the newly introduced subclasses.

Geometric Properties of Generalized Integral Operators Related to The Miller–Ross Function

It is very well-known that the special functions and integral operators play a vital role in the research of applied and mathematical sciences. In this paper, our aim is to present sufficient conditions for the families of integral operators containing the normalized forms of the Miller–Ross functions such that they can be univalent in the open unit disk. Moreover, we find the convexity order of these operators. In proof of results, we use some differential inequalities related with Miller–Ross functions and well-known lemmas. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples.

Risk aggregation with FGM copulas

We offer a new perspective on risk aggregation with FGM copulas. Along the way, we discover new results and revisit existing ones, providing simpler formulas than one can find in the existing literature. This paper builds on two novel representations of FGM copulas based on symmetric multivariate Bernoulli distributions and order statistics. First, we detail families of multivariate distributions with closed-form solutions for the cumulative distribution function or moments of the aggregate random variables. We provide methods to compute the cumulative distribution function of aggregate rvs when the marginals are discrete, then order aggregate random variables under the convex order. Finally, we discuss risk-sharing and capital allocation, providing numerical examples for each.

A Model for Risk Adjustment (IFRS 17) for Surrender Risk in Life Insurance

We propose a model for risk adjustment, in the context of IFRS 17, for surrender risk. Surrender rates are assumed to follow a stochastic process, underpinned by data. The distribution of the present value of future individual cash flows is calculated. Using well-known techniques from the theory of convex ordering of stochastic variables, we present closed formula approximations of risk measures, such as quantiles, for the total portfolio. These formulas are easy to program and enable an insurance company to calculate its risk adjustment without time-consuming simulations.

Starlike Functions Associated with Secant Hyperbolic Function

Motivated by the recent work on the symmetric domains, this article investigates certain features of symmetric domain which are caused by the secant hyperbolic functions. Geometric characteristics of analytic functions associated with secant hyperbolic functions are discussed, which include the inclusion results, structural formula, certain sharp radii results such as radius of starlikeness and convexity of order α. It also finds a radius for ratios of analytic functions associated with Euler numbers.

Applications of Strassen’s theorem and Choquet theory to optimal transport problems, to uniformly convex functions and to uniformly smooth functions

We provide a unifying interpretation of various optimal transport problems as a minimisation of a linear functional over the set of all Choquet representations of a given pair of probability measures ordered with respect to a certain convex cone of functions. This allows us to provide novel proofs of duality formulae. Among our tools is Strassen’s theorem.We provide new formulations of the primal and the dual problem for martingale optimal transport employing a novel representation of the set of extreme points of probability measures in convex order on Euclidean space. We exhibit a link to uniformly convex and uniformly smooth functions and provide a new characterisation of such functions.We introduce a notion of martingale triangle inequality. We show that Kantorovich–Rubinstein duality bears an analogy in the martingale setting employing the cost functions that satisfy the inequality.

On the Integral Representation and the Raşa, Jensen and Hermite–Hadamard Inequalities for Box-Convex Functions

We give the integral representation of Popoviciu’s (Mathematica 8:1–85, 1934) box-(m, n)-convex functions. Based on this integral representation, we obtain a characterization of box-(m, n)-convex orders, which we then use in the proofs of the Raşa type, the Hermite–Hadamard type and the Jensen inequalities for box-(m, n)-convex functions as well as for strongly box-(m, n)-convex functions. We extend the results obtained by Gal and Niculescu (J Math Anal Appl 479:903–925, 2019), and by Gavrea and Gavrea (Mediterr J Math 19:18–33, 2022).

On the Analytical Determination of Geometric Characterizations of Analytic Functions

As it is known, there are many sufficient conditions for the classification complex functions of one variable f(z), which are analytic and univalent in the open unit disc U = {z ∈ C ∶ SzS < 1}, and are also normalized with f(0) = 1 − f′(0) = 0 which are also known as normalization conditions. In this sense, the main goal of present article is to derive some special sufficient conditions for f(z) to be starlike of order 2−r and convex of order 2−r in U , with r is a positive integer.

Generalized location-scale mixtures of elliptical distributions: Definitions and stochastic comparisons

This article proposes a unified class of generalized location-scale mixture of multivariate elliptical distributions and studies integral stochastic orderings of random vectors following such distributions. Given a random vector Z , independent of X and Y , the scale parameter of this class of distributions is mixed with a function α ( Z ) and its skew parameter is mixed with another function β ( Z ) . Sufficient (and necessary) conditions are established for stochastically comparing different random vectors stemming from this class of distributions by means of several stochastic orders including the usual stochastic order, convex order, increasing convex order, supermodular order, and some related linear orders. Two insightful assumptions for the density generators of elliptical distributions, aiming to control the generators’ tail, are provided to make stochastic comparisons among mixed-elliptical vectors. Some applications in applied probability and actuarial science are also provided as illustrations on the main findings.

Lipschitz continuity of the Wasserstein projections in the convex order on the line

Wasserstein projections in the convex order were first considered in the framework of weak optimal transport, and found applications in various problems such as concentration inequalities and martingale optimal transport. In dimension one, it is well-known that the set of probability measures with a given mean is a lattice w.r.t. the convex order. Our main result is that, contrary to the minimum and maximum in the convex order, the Wasserstein projections are Lipschitz continuity w.r.t. the Wasserstein distance in dimension one. Moreover, we provide examples that show sharpness of the obtained bounds for the 1-Wasserstein distance.

On the near-viability property of controlled mean-field flows

We aim at studying the property of controlled stochastic flows with mean-field dynamics to comply with some (closed) state restrictions. This property, known as (near)-viability, is tackled via (quasi-)tangency methods. Law restrictions and mixed state-law restrictions are considered as the interplay between the two classes. As an auxiliary result used in this process, Theorem 1.2, whose importance exceeds the present framework, dissociates, through a class of elementary controls, the contribution of the Brownian filtration and that of the initial $ \sigma $-field. Explicit conditions for the coefficient functions are provided in the invariance context. Moreover, specific applications to comparison in the convex order illustrate the theoretical results.

Subordination Properties of Certain Operators Concerning Fractional Integral and Libera Integral Operator

The results contained in this paper are the result of a study regarding fractional calculus combined with the classical theory of differential subordination established for analytic complex valued functions. A new operator is introduced by applying the Libera integral operator and fractional integral of order λ for analytic functions. Many subordination properties are obtained for this newly defined operator by using famous lemmas proved by important scientists concerned with geometric function theory, such as Eenigenburg, Hallenbeck, Miller, Mocanu, Nunokawa, Reade, Ruscheweyh and Suffridge. Results regarding strong starlikeness and convexity of order α are also discussed, and an example shows how the outcome of the research can be applied.

Normalized generalized Bessel function and its geometric properties

The normalization of the generalized Bessel functions mathrm{U}_{sigma,r}(sigma,rin mathbb{C}mathbbm{)} defined by Uσ,r(z)=z+∑j=1∞(−r)j4j(1)j(σ)jzj+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} \\mathrm{U}_{\\sigma,r}(z)=z+\\sum_{j=1}^{\\infty} \\frac{(-r)^{j}}{4^{j} (1)_{j}(\\sigma )_{j}}z^{j+1} \\end{aligned}$$ \\end{document} was introduced, and some of its geometric properties have been presented previously. The main purpose of the present paper is to complete the results given in the literature by employing a new procedure. We first used an identity for the logarithmic of the gamma function as well as an inequality for the digamma function to establish sufficient conditions on the parameters so that mathrm{U}_{sigma,r} is starlike or convex of order α(0leq alpha leq 1) in the open unit disk. Moreover, the starlikeness and convexity of mathrm{U}_{sigma,r} have been considered where the leading concept of the proofs comes from the starlikeness of the power series f(z)=sum_{j=1}^{infty}A_{j}z^{j} and the classical Alexander theorem between the classes of starlike and convex functions. We gave a simple proof to show that our conditions are not contradictory. Ultimately, the close-to-convexity of (zcos sqrt{z} ) ast mathrm{U}_{sigma,r} and (sin z ) ast frac {mathrm{U}_{sigma,r}(z^{2})}{z} have been determined, where “∗” stands for the convolution between the power series.

Pareto-optimal reinsurance for both the insurer and the reinsurer under the risk-adjusted value and general premium principles

In this paper, we design the Pareto-optimal reinsurance contract for both the insurer and the reinsurer by minimizing the convex combination of the risk-adjusted value of the insurer’s liability and the reinsurer’s liability, where capital at risk is calculated by the value at risk (VaR) or conditional value at risk (CVaR). In order to prevent the moral hazard, we assume that both ceded and retained loss functions are increasing functions. We analyze the optimal solutions for a wide class of reinsurance premium principles. When the reinsurance premium principles satisfy three axioms: law invariance, risk loading and preserving convex order, we find that layer reinsurance is always optimal over the assumed risk measures. Then we impose an additional weak constraint on the premium principle to simplify the form of layer reinsurance which is optimal. Finally, we illustrate the applicability of our results by deriving the parameters of the optimal layer reinsurance explicitly under the expected value principle and Wang’s premium principle.

From risk reduction to risk elimination by conditional mean risk sharing of independent losses

This paper studies diversification effects resulting from pooling insurance losses according to the risk allocation rule proposed by Denuit and Dhaene (2012). General comparison results are established for conditional expectations given sums of independent random variables. It is shown that these expectations decrease in the number of terms comprised in the conditioning sums. Additional inequalities are obtained under regression dependence in the sum. These general results are used to derive the monotonicity of the respective contributions of the participants with respect to the convex order, showing that increasing the number of participants is always beneficial under conditional mean risk sharing. New convergence results are obtained, showing that the variance of individual contributions tends to zero in many interesting cases. This provides actuaries with conditions ensuring that the risk can be fully eliminated within the pool, at the limit.

A Moment Inequality for the NBRULC Class: Statistical Properties with Applications to Model Asymmetric Data

In this paper, the moment inequalities for some aging distributions are derived based on a mathematical class entitled “a new better than renewal used in Laplace transform order in increasing convex order class (NBRULC)”. The introduced inequalities can be utilized as a new mathematical test for the exponentiality property versus NBRULC. If the mean life is finitely based on these inequalities, then all higher-order moments exist. Pitman’s asymptotic efficiency of the new mathematical test is derived and studied in detail for some asymmetric probability models. The new mathematical test’s power is estimated in reliability studies for a few well-known alternative asymmetric models. The problem in the case of right-censored data is also handled. After that, applying the suggested test to practical issues is demonstrated using asymmetric, real datasets.

Representative endowments and uniform Gini orderings of multi-attribute welfare

For the comparison of inequality and welfare in multiple attributes the use of generalized Gini indices is proposed. Individual endowment vectors are summarized by using attribute weights and aggregated in a spectral social evaluation function. Such functions are based on classes of spectral functions, ordered by their aversion to inequality. Given a spectrum and a set P of attribute weights, a multivariate Gini dominance ordering, being uniform in weights, is defined. If the endowment vectors are comonotonic, the dominance is determined by their marginal distributions; if not, the dependence structure of the endowment distribution has to be taken into account. For this, a set-valued representative endowment is introduced that characterizes the welfare of a d-dimensioned distribution. It consists of all points above the lower border of a convex compact in mathbb {R}^{d}, while the set ordering of representative endowments corresponds to uniform Gini dominance. An application is given to the welfare of 28 European countries. Properties of P-uniform Gini dominance are derived, including relations to other orderings of d-variate distributions such as convex and dependence orderings. The multi-dimensioned representative endowment can be efficiently calculated from data. In a sampling context, it consistently estimates its population version.

Some Results on Measures of Interaction among Risks

It has become a common understanding that financial risk can spread rapidly from one institution to another, and the stressful status of one institution may finally result in a systemic crisis. One popular method to assess and quantify the risk of contagion is employing the co-risk measures and risk contribution measures. It is interesting and important to understand how the underlining dependence structure and magnitude of random risks jointly affect systemic risk measures. In this paper, we mainly focus on the conditional value-at-risk, conditional expected shortfall, the delta conditional value-at-risk, and the delta conditional expected shortfall. Existing studies mainly focus on the situation with two random risks, and this paper makes some contributions by considering the scenario with possibly more than two random risks. By employing the tools of stochastic order, positive dependence concepts and arrangement monotonicity, several results concerning the usual stochastic order, increasing convex order, dispersive order and excess wealth order are presented. Concisely speaking, it is found that for a large enough stress level, a larger random risk tends to lead to a more severe systemic risk. We also performed some Monte Carlo experiments as illustrations for the theoretical findings.