Two Sufficient Conditions for Convex Ordering on Risk Aggregation

Abstract

We define new stochastic orders in higher dimensions called weak correlation orders. It is shown that weak correlation orders imply stop-loss order of sums of multivariate dependent risks with the same marginals. Moreover, some properties and relations of stochastic orders are discussed.

1. Introduction

Correlation order as an important stochastic order relation was first introduced by Joe [1]; Dhaene and Goovaerts [2] studied the bivariate case with the same marginals. After that, the bivariate case has been generalized by Lu and Zhang [3]. Recall that given two random vectors and with the same marginals, is said to be less correlated than , written as , if for every pair of disjoint subsets and of ,where and are nondecreasing functions for which the covariances exist. The main result of Lu and Zhang [3] showed that the correlation order implied stop-loss order for portfolios of multivariate dependent risks; that is, implies . Stop-loss order as a special case of convex order is the most frequently used order relation for the comparison of risks, written as , for any two random variables and , if and only if the inequality holds for all real , where denotes the positive part of the real . In addition, is said to precede in the convex order sense; define , if and only if and More details and other characterizations about stop-loss order can be found in Denuit et al. [4], Dhaene et al. [5], Landsman and Tsanakas [6], and Shaked and Shanthikumar [7]. Rüschendorf [8] introduced a new dependence order among risks called the weakly conditional increasing in sequence order; by definition given two random vectors and with the same marginals, is said to be smaller than in the weakly conditional increasing in sequence order, written as , if for all , all and is monotonically nondecreasing,where and . It is showed that more positive dependence with respect to the wcs ordering implied more risk with respect to the supermodular ordering (see Rüschendorf [8] for the definitions of supermodular function and supermodular ordering) for -dimensional random vectors; that is, implies . Note that, in (1), we let and , (2) is an immediate consequence of (1), and by Example 5 of this paper, the weakly conditional increasing in sequence order is weaker than correlation order, while the stop-loss order still holds by Müller [9]. Enlightened by this, we are committed to find more general conditions for the multivariate case which can also imply stop-loss order.

In this short note, we give the concepts of weak correlation orders in higher dimensions and show that the weak correlation orders imply stop-loss order of multivariate dependent risks with the same marginals. The remainder of the paper is organized as follows. In Section 2, we introduce some concepts of stochastic orders including the new definitions and discuss the properties and stochastic order relations. The main results of this paper are presented and proved in Section 3.

2. Preliminaries

Given Fréchet space of all -dimensional random vectors , we have as marginal distributions, where , and the joint distribution function is For all , we have the following inequality: where and are called Fréchet upper bound and Fréchet lower bound of , respectively. Remark that is reachable in and when , is indeed a distribution function. However, when , is no longer always a distribution function (see Denuit et al. [4]). A necessary and sufficient condition for to be a distribution function in can be found in Dhaene and Denuit [10]. Throughout the short note, it is assumed that all random variables are real random variables on this space.

Comparing random variables is the essence of the actuarial profession; in order to acquire more general results, we give the notion of weak correlation orders as follows.

Definition 1. Let random vectors and be elements of , and we say that is smaller than in type I weak correlation order, written as , if for all , , any of the following equivalent conditions holds:where is an indicator function.

Remark 2. The equivalence between (5) can be obtained as follows. We haveand by the same way, we obtain Hence, (5) are equivalent. Moreover, (8) and (9) are also equivalent.

Definition 3. Random vectors and are elements of , and we say that is smaller than in type II weak correlation order, written as , if for all , , any of the following equivalent conditions holds:

Remark 4. Obviously, (8) can be derived from (2) by letting

Let random vectors and be elements of with values in , is the distribution function, and is the survival function of ; define , iff , for any ; , iff , for any ; , iff , for all supermodular functions such that the expectations exist. For more discussion about supermodular order (sm order) and orthant orders (uo order and lo order), see Shaked and Shanthikumar [11], Denuit and Mesfioui [12], and Kzldemir and Privault [13]. The following relations are well known by the definitions above, Lu and Zhang [3] and Müller [9]:

(i) and ;

(ii) ;

(iii) .

Specially, these stochastic orders are equivalent in the bivariate case.

It is well known that correlated order and weakly conditional increasing in sequence order can deduce the weak correlation orders, but the converse is not true in case ; the following example illustrates this point.

Example 5. Let and be random valuables with distributions as , , , and . In addition, and are random vectors with the following joint distributions: For all , it is easy to get that However, for two nondecreasing functions and , we haveThis is becauseIn the same way, we can getThen we obtain and
Let , so is nondecreasing; then we haveSimilarly,so cannot deduce .

The next property is straightforward; we omitted all the minor details.

Property 6. Let two random vectors and be elements of , for any independent random vector which is independent of and ; if , we haveand if , we can get

3. Main Results and Proofs

In this section, we will give the main results of this paper. That the correlation order implies stop-loss order for portfolios of multivariate dependent risks has been investigated by Lu and Zhang [3] and Zhang and Weng [14]. In the next theorems, we will show that the property still holds in weak correlation orders.

Theorem 7. Let and be elements of , and if , then

To prove Theorem 7, we need the following two lemmas. For simplicity, let random variables marked with asterisks be independent, and every random variable marked with asterisk has the same distribution with its original.

Lemma 8. For any random vector and for all , the following equality holds:

Proof. For any , we have so that

Lemma 9 (Shaked and Shanthikumar [7]). If random variables satisfy that , and have the same distribution and are independent of and , then .

Proof of Theorem 7. For bivariate case, by Lemma 8 we haveAssume that it is true for that we will prove that it is also true for in the following. Define symbols , and , , and from Lemma 8, we haveso thatSince and have the same distribution, by induction and Lemma 9, we obtain Hence, we finish the proof.

The next theorem shows that still holds in -II order.

Theorem 10. Let and be elements of ; if , then

Proof. The proof follows immediately by the same method as Theorem 7 and Lemmas 8 and 9.

Remark 11. In fact, since , it follows from Theorem 7 that

Remark 12. If is a random vector in , such that or for all , then is comonotonic. Recall the definition of comonotonic: is said to be comonotonic, if for any So is reachable in

Remark 13. Let and be the essential infimum and essential supremum of a random variable , is a fixed random vector in which satisfies or , and if or for all , then is mutually exclusive.

More details about comonotonicity and mutual exclusivity can be found in Dhaene and Denuit [10], Cheung and Lo [15, 16], Mesfioui and Denuit [17], and Puccetti and Wang [18].

Corollary 14. Let random vectors and be elements of , random variable is independent of and , and if or holds, then

Proof. The proof can be obtained immediately by Lemma 9 and Theorems 7 and 10.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The research was supported by the National Natural Science Foundation of China (nos. 11571198 and 11501319), Project Funded by China Postdoctoral Science Foundation (no. 2015M582064), and the Natural Science Foundation of Shandong (nos. ZR2015AL013 and ZR2014AM021).