Convex Order Research Articles

One Dimensional Martingale Rearrangement Couplings

We are interested in martingale rearrangement couplings. As introduced by Wiesel in order to prove the stability of Martingale Optimal Transport problems, these are projections in adapted Wasserstein distance of couplings between two probability measures on the real line in the convex order onto the set of martingale couplings between these two marginals. In reason of the lack of relative compactness of the set of couplings with given marginals for the adapted Wasserstein topology, the existence of such a projection is not clear at all. Under a barycentre dispersion assumption on the original coupling which is in particular satisfied by the Hoeffding-Frechet or comonotone coupling, Wiesel gives a clear algorithmic construction of a martingale rearrangement when the marginals are finitely supported and then gets rid of the finite support assumption by relying on a rather messy limiting procedure to overcome the lack of relative compactness. Here, we give a direct general construction of a martingale rearrangement coupling under the barycentre dispersion assumption. This martingale rearrangement is obtained from the original coupling by an approach similar to the construction we gave in Jourdain and Margheriti [Electr. J. Probab. (2020)] of the inverse transform martingale coupling, a member of a family of martingale couplings close to the Hoeffding-Fréchet coupling, but for a slightly different injection in the set of extended couplings introduced by Beiglböck and Juillet and which involve the uniform distribution on [0,1] in addition to the two marginals. We last discuss the stability in adapted Wassertein distance of the inverse transform martingale coupling with respect to the marginal distributions.

Shadow martingales – a stochastic mass transport approach to the peacock problem

Given a family of real probability measures $(\mu_t)_{t\geq 0}$ increasing in convex order (a peacock) we describe a systematic method to create a martingale exactly fitting the marginals at any time. The key object for our approach is the obstructed shadow of a measure in a peacock, a generalization of the (obstructed) shadow introduced in \cite{BeJu16,NuStTa17}. As input data we take an increasing family of measures $(\nu^\alpha)_{\alpha \in [0,1]}$ with $\nu^\alpha(\mathbb{R})=\alpha$ that are submeasures of $\mu _0$, called a parametrization of $\mu_0$. Then, for any $\alpha$ we define an evolution $(\eta^\alpha_t)_{t\geq 0}$ of the measure $\nu^\alpha=\eta^\alpha_0$ across our peacock by setting $\eta^\alpha_t$ equal to the obstructed shadow of $\nu^\alpha$ in $(\mu _s)_{s \in [0,t]}$. We identify conditions on the parametrization $(\nu^\alpha)_{\alpha\in [0,1]}$ such that this construction leads to a unique martingale measure $\pi$, the shadow martingale, without any assumptions on the peacock. In the case of the left-curtain parametrization $(\nu_{\text{lc}}^\alpha)_{\alpha \in [0,1]}$ we identify the shadow martingale as the unique solution to a continuous-time version of the martingale optimal transport problem. Furthermore, our method enriches the knowledge on the Predictable Representation Property (PRP) since any shadow martingale comes with a canonical Choquet representation in extremal Markov martingales.

The potential of the shadow measure

It is well known that given two probability measures μ and ν on R in convex order there exists a discrete-time martingale with these marginals. Several solutions are known (for example from the literature on the Skorokhod embedding problem in Brownian motion). But, if we add a requirement that the martingale should minimise the expected value of some functional of its starting and finishing positions then the problem becomes more difficult. Beiglböck and Juillet (Ann. Probab. 44 (2016) 42–106) introduced the shadow measure which induces a family of martingale couplings, and solves the optimal martingale transport problem for a class of bivariate objective functions. In this article we extend their (existence and uniqueness) results by providing an explicit construction of the shadow measure and, as an application, give a simple proof of its associativity.

On q‐Convex Functions Defined by the q‐Ruscheweyh Derivative Operator in Conic Regions

The core objective of this article is to introduce and investigate a new class of convex functions associated with the conic domain defined by the Ruscheweyh q‐differential operator. Many interesting properties such as sufficiency criteria, coefficient bounds, partial sums, and radius of convexity of order α for the functions of the said class are investigated here.

On endomorphisms of the inverse semigroup of convex order isomorphisms of a bounded rank which are generated by Rees congruences

On endomorphisms of the inverse semigroup of convex order isomorphisms of a bounded rank which are generated by Rees congruences

Diffusion-limited annihilating systems and the increasing convex order

We consider diffusion-limited annihilating systems with mobile A-particles and stationary B-particles placed throughout a graph. Mutual annihilation occurs whenever an A-particle meets a B-particle. Such systems, when ran in discrete time, are also referred to as parking processes. We show for a broad family of graphs and random walk kernels that augmenting either the size or variability of the initial placements of particles increases the total occupation time by A-particles of a given subset of the graph. A corollary is that the same phenomenon occurs with the total lifespan of all particles in internal diffusion-limited aggregation.

The Markov-quantile process attached to a family of marginals

Let µ = (µt)t∈R be any 1-parameter family of probability measures on R. Its quantile process (Gt)t∈R : ]0, 1[ → RR, given by Gt(α) = inf{x ∈ R : µt(]−∞, x]) > α}, is not Markov in general. We modify it to build the Markov process we call “Markov-quantile”. We first describe the discrete analogue: if (µn)n∈Z is a family of probability measures on R, a Markov process Y = (Yn)n∈Z such that Law(Yn) = µn is given by the data of its couplings from n to n + 1, i.e. Law((Yn, Yn+1)), and the process Y is the inhomogeneous Markov chain having those couplings as transitions. Therefore, there is a canonical Markov process with marginals µn and as similar as possible to the quantile process: the chain whose transitions are the quantile couplings. We show that an analogous process exists for a continuous parameter t: there is a unique Markov process X with the measures µt as marginals, and being a limit for the finite dimensional topology of quantile processes where the past is made independent of the future at finitely many times (many non-Markovian limits exist in general). The striking fact is that the construction requires no regularity for the family µ. We rely on order arguments, which seems to be completely new for the purpose. We also prove new results the Markov-quantile process yields in two contemporary frameworks: – In case µ is increasing for the stochastic order, X has increasing trajectories. This is an analogue of a result of Kellerer dealing with the convex order, peacocks and martingales. Modifiying Kellerer’s proof, we also prove simultaneously his result and ours in this case. – If µ is absolutely continuous in Wasserstein space P2(R) then X is solution of a Benamou–Brenier transport problem with marginals µt. It provides a Markov probabilistic representation of the continuity equation, unique in a certain sense.

Risk aggregation under dependence uncertainty and an order constraint

We study the aggregation of two risks when the marginal distributions are known and the dependence structure is unknown, under the additional constraint that one risk is smaller than or equal to the other. Risk aggregation problems with the order constraint are closely related to the recently introduced notion of the directional lower (DL) coupling. The largest aggregate risk in concave order (thus, the smallest aggregate risk in convex order) is attained by the DL coupling. These results are further generalized to calculate the best-case and worst-case values of tail risk measures. In particular, we obtain analytical formulas for bounds on Value-at-Risk. Our numerical results suggest that the new bounds on risk measures with the extra order constraint can greatly improve those with full dependence uncertainty.

Global boundedness of functions of finite order that are bounded outside small sets

We prove that subharmonic or holomorphic functions of finite order on the plane, in space, or on the unit disc or ball that are bounded above on a sequence of circles or spheres, or on a system of embedded discs or balls, outside some asymptotically small sets are bounded above throughout. Hence, subharmonic functions of finite order on the complex plane, entire or plurisubharmonic functions of finite order, and also convex or harmonic functions of finite order that are bounded above on spheres outside such sets are constants. The results and the approaches to the proofs are new for both functions of one and several variables. Bibliography: 14 titles.

Hessian and increasing-Hessian orderings of scale-shape mixtures of multivariate skew-normal distributions and applications

In this work, we establish some stochastic comparison results for the class of scale-shape mixtures of multivariate skew-normal (SSSN) distributions. These multivariate stochastic comparisons involve Hessian and increasing-Hessian orderings as well as many of their special cases. We show that the supermodular and increasing-supermodular orderings are equivalent to the concordance and upper orthant orderings, respectively, as well as the ordering of correlations within the underlying vectors of variables. The supermodular and increasing-supermodular orderings of log-SSSN family are also shown to be equivalent to the convex ordering and increasing-convex orderings of their geometric and harmonic averages. These results are then used to order the stability of average of relatives and aggregate value of portfolios through the order of their correlations. The aggregate claims of insurance portfolios are ordered, in the sense of stop-loss ordering, through the order of their average correlations. A multivariate stop-loss order is proposed to order the multivariate aggregate claims in SSSN family and is then shown to be equivalent to the concordance and upper orthant orderings in the sense of supermodular and increasing-supermodular orderings, respectively, based on the ordering of interdependence of claims in the portfolios. In addition, the Gini index of SSSN risks is ordered, in the sense of increasing-convex order, with respect to the strength of correlations. Finally, the results are used in analyzing some real data sets to provide illustrations for the established results and practical interpretations.

Enhancing an insurer's expected value by reinsurance and external financing

In this paper, we analyze a decision-making problem for an insurer with limited liability, who is subject to a solvency constraint and wants to maximize the expected value through reinsurance purchase and external financing. We impose mild conditions on the reinsurance premium principle, which are the axioms of law invariance, risk loading and preserving the convex order. These three axioms are satisfied by all the widely used premium principles, except the Esscher principle, listed in Young (2004). If value at risk (VaR) or conditional value at risk (CVaR) is adopted to calculate the insurer's regulatory capital, we show that the optimal reinsurance can be in the form of two layers. The optimal reinsurance form is reduced to one layer once the premium principle further satisfies a weak condition. In particular, under Wang's, expected value, variance and Dutch premium principles, we derive the insurer's optimal strategies of reinsurance and financing, which are obtained explicitly under the VaR risk measure but have to be solved numerically for the CVaR risk measure. Results indicate that the insurer has three types of optimal strategies: the reinsurance only strategy, external financing only strategy, and mixed strategy. Moreover, the insurer's attitude toward ceding the risk through reinsurance and increasing capital through external financing is often greatly affected by the regulatory regime, the financing cost and the reinsurance price.

Stochastic comparisons on extreme order statistics from observations associated by FGM copula

This study deals with the stochastic comparison of extreme order statistics from random samples with FGM copula or FGM survival copula. For general marginal distributions, several sufficient conditions on the usual stochastic order, the reversed hazard rate order, the hazard rate order, the increasing convex order and the increasing concave order on sample maximums or minimums are derived. Then, we apply these conditions to samples following proportional hazard rate model, proportional reversed hazard rate model, Weibull distributions and negative binomial distributions, and generalize some ordering results about samples having independent observations to the case with FGM copula or FGM survival copula. Two comparing results on the second-order statistics from proportional hazard samples are presented. Some numerical examples are provided to illustrate the theoretical results as well.

Mapping properties of the zero-balanced hypergeometric functions

In this paper, the order of convexity of z2F1(a,b;c;z) is given under certain conditions on the positive real parameters a, b and c. We show also that the image domains of the unit disc D under some shifted zero-balanced hypergeometric functions z2F1(a,b;a+b;z) are convex and bounded by two horizontal lines. This solves a problem raised by Ponnusamy and Vuorinen in [10].

Correspondence between a new class of generalized cone convexity and higher order duality

Correspondence between a new class of generalized cone convexity and higher order duality

Stochastic Order and Generalized Weighted Mean Invariance.

In this paper, we present order invariance theoretical results for weighted quasi-arithmetic means of a monotonic series of numbers. The quasi-arithmetic mean, or Kolmogorov–Nagumo mean, generalizes the classical mean and appears in many disciplines, from information theory to physics, from economics to traffic flow. Stochastic orders are defined on weights (or equivalently, discrete probability distributions). They were introduced to study risk in economics and decision theory, and recently have found utility in Monte Carlo techniques and in image processing. We show in this paper that, if two distributions of weights are ordered under first stochastic order, then for any monotonic series of numbers their weighted quasi-arithmetic means share the same order. This means for instance that arithmetic and harmonic mean for two different distributions of weights always have to be aligned if the weights are stochastically ordered, this is, either both means increase or both decrease. We explore the invariance properties when convex (concave) functions define both the quasi-arithmetic mean and the series of numbers, we show its relationship with increasing concave order and increasing convex order, and we observe the important role played by a new defined mirror property of stochastic orders. We also give some applications to entropy and cross-entropy and present an example of multiple importance sampling Monte Carlo technique that illustrates the usefulness and transversality of our approach. Invariance theorems are useful when a system is represented by a set of quasi-arithmetic means and we want to change the distribution of weights so that all means evolve in the same direction.

Peacock geodesics in Wasserstein space

Martingale optimal transport has attracted much attention due to its application in pricing and hedging in mathematical finance. The essential notion which makes martingale optimal transport different from optimal transport is peacock. A peacock is a sequence of measures which satisfies convex order property. In this paper we study peacock geodesics in Wasserstain space, and we hope this paper can provide some geometrical points of view to look at martingale optimal transport.

K-combined random fields: Basic properties and stochastic orderings

This paper introduces the K-combined vector random field, whose finite-dimensional characteristic functions are made up of certain power functions and whose finite-dimensional density functions are comprised of the modified Bessel functions of the second type. It is an elliptically contoured vector random field, contains K-differenced vector random field of Alsultan and Ma (2019) as a special case, and possesses all orders of moments. It is fully characterized by its mean vector function and its covariance matrix function, just like a Gaussian vector random field. With various selection of its parameters, its finite-dimensional distributions may have heavy tails or thin tails, comparing with a Gaussian one, and thus it provides a potential model for applications. We also investigate the usual stochastic ordering, the convex ordering, and the peakedness ordering of K-combined random fields and of multivariate K-combined distributions, with necessary and/or sufficient conditions derived.

Weekly scheduling of emergency department physicians to cope with time-varying demand

Overcrowding, long waiting times and delays frequently occur in hospital Emergency Departments (EDs). The main causes are the stochastic and strongly time-varying demands of patient arrivals at an ED and the temporary overloading of EDs. Motivated by collaboration with large EDs, we investigate the physicians scheduling problem in the ED for a weekly planning horizon to address the stochastic and time-varying demands. The patient–physician service system is modeled as a time-varying and temporarily overloaded queueing system without abandonments. We employ a continuous-time Markov chain and uniformization method for the analytical evaluation of waiting times of patients. Based on an increasing convex order property, patient waiting times are proven to be convex in a system state. Based on this convexity, an approximation technique is established to model the physician scheduling problem as a mixed-integer program to decide the start and end working times of physicians. We also obtain a tight lower bound of the optimal solution to this scheduling problem. A local search-based algorithm is designed to solve this scheduling problem. Our method improves the physician schedule obtained via the approaches from the literature, significantly improves actual hospital scheduling, and simultaneously reduces physician working times and patient waiting times.

OPTIMAL REINSURANCE FROM THE VIEWPOINTS OF BOTH AN INSURER AND A REINSURER UNDER THE CVAR RISK MEASURE AND VAJDA CONDITION

ABSTRACTIn this paper, we study the optimal reinsurance contracts that minimize the convex combination of the Conditional Value-at-Risk (CVaR) of the insurer’s loss and the reinsurer’s loss over the class of ceded loss functions such that the retained loss function is increasing and the ceded loss function satisfies Vajda condition. Among a general class of reinsurance premium principles that satisfy the properties of risk loading and convex order preserving, the optimal solutions are obtained. Our results show that the optimal ceded loss functions are in the form of five interconnected segments for general reinsurance premium principles, and they can be further simplified to four interconnected segments if more properties are added to reinsurance premium principles. Finally, we derive optimal parameters for the expected value premium principle and give a numerical study to analyze the impact of the weighting factor on the optimal reinsurance.

On the Order of Convexity for the Shifted Hypergeometric Functions

In the present paper, we study the order of convexity of $$z{_2F_1}(a,b;c;z)$$ with real parameters a, b and c where $${_2F_1}(a,b;c;z)$$ is the Gaussian hypergeometric function. First we obtain some conditions for $$z{_2F_1}(a,b;c;z)$$ with no finite orders of convexity by considering its asymptotic behavior around $$z=1$$ . Then the order of convexity of $$z{_2F_1}(a,b;c;z)$$ is demonstrated for some ranges of real parameters a, b and c. In the last section, we give some examples as applications of the main results.