In this paper, the ordering properties of convex and increasing convex orders of the dependent random variables are studied. Some closure properties of the convex and increasing convex orders under independent random variables are extended to the dependent random variables under the Archimedean copula. Two applications are provided to illustrate our results.

In this short note we introduce two notions of dispersion-type variability orders, namely expected shortfall-dispersive (ES-dispersive) order and expectile-dispersive (ex-dispersive) order, which are defined by two classes of popular risk measures, the expected shortfall and the expectiles. These new orders can be used to compare the variability of two risk random variables. It is shown that either the ES-dispersive order or the ex-dispersive order is the same as the dilation order. This gives us some insight into parametric measures of variability induced by risk measures in the literature.

The class of distortion riskmetrics is defined through signed Choquet integrals, and it includes many classic risk measures, deviation measures, and other functionals in the literature of finance and actuarial science. We obtain characterization, finiteness, convexity, and continuity results on general model spaces, extending various results in the existing literature on distortion risk measures and signed Choquet integrals. This paper offers a comprehensive toolkit of theoretical results on distortion riskmetrics which are ready for use in applications.

A retrospective rating plan, whose insurance premium depends upon an insured's actual loss during the policy period, is a special insurance agreement widely used in liability insurance. In this paper, the design of an optimal retrospective rating plan is analyzed from the perspective of the insured who seeks to minimize its risk exposure in the sense of convex order. In order to reduce the moral hazard, we assume that both the insured and the insurer are obligated to pay more for a larger realization of the loss. Under the further assumptions that the minimum premium is zero, the maximum premium is proportional to the expected indemnity, and the basic premium is the only free parameter in the formula for retrospective premium given by Meyers (2004) and that the basic premium is determined in such a way that the expected retrospective premium equates to the expected indemnity with a positive safety loading, we formally establish the relationship that the insured will suffer more risk for a larger loss conversion factor or a higher maximum premium. These findings suggest that the insured prefers an insurance policy with the expected value premium principle, which is a special retrospective premium principle with zero loss conversion factor. In addition, we show that any admissible retrospective rating plan is dominated by a stop-loss insurance policy. Finally, the optimal retention of a stop-loss insurance is derived numerically under the criterion of minimizing the risk-adjusted value of the insured's liability where the liability valuation is carried out using the cost-of-capital approach based on the conditional value at risk.

We consider the problem of stochastic comparison of general GARCH-like processes for different parameters and different distributions of the innovations. We identify several stochastic orders that are propagated from the innovations to the GARCH process itself, and we discuss their interpretations. We focus on the convex order and show that in the case of symmetric innovations it is also propagated to the cumulated sums of the GARCH process. More generally, we discuss multivariate comparison results related to the multivariate convex and supermodular orders. Finally, we discuss ordering with respect to the parameters in the GARCH(1, 1) case.

We use multivariate total positivity theory to exhibit new families of peacocks. As theauthors of [F. Hirsch, C. Profeta, B. Roynette and M. Yor, Peacocks and associatedmartingales vol. 3. Bocconi-Springer (2011)], our guiding example is the resultof Carr−Ewald−Xiao [P. Carr, C.-O. Ewald and Y. Xiao,Finance Res. Lett. 5 (2008) 162–171]. We shall introducethe notion of strong conditional monotonicity. This concept is strictly more restrictivethan the conditional monotonicity as defined in [F. Hirsch, C. Profeta, B. Roynette and M.Yor, Peacocks and associated martingales, vol. 3. Bocconi-Springer (2011)] (see also [R.H.Berk, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42(1978) 303–307], [A.M. Bogso, C. Profeta and B. Roynette, Lect. Notes Math.Springer, Berlin (2012) 281–315.] and [M. Shaked and J.G. Shanthikumar,Probab. Math. Statistics. Academic Press, Boston (1994)].). There aremany random vectors which are strongly conditionally monotone (SCM). Indeed, we shallprove that multivariate totally positive of order 2 (MTP2) random vectors are SCM. As aconsequence, stochastic processes with MTP2 finite-dimensional marginals are SCM. This familyincludes processes with independent and log-concave increments, and one-dimensionaldiffusions which have absolutely continuous transition kernels.

An optimal reinsurance problem from the perspective of an insurer is studied in this paper, where an upper limit is imposed on a reinsurer's expected loss over a prescribed level. In order to reduce the moral hazard, we assume that both the insurer and the reinsurer are obligated to pay more as the amount of loss increases in a typical reinsurance treaty. We further assume that the optimization criterion preserves the convex order. Such a criterion is very general as most of the criteria for optimal reinsurance problems in the literature preserve the convex order. When the reinsurance premium is calculated as a function of the actuarial value of coverage, we show via a stochastic dominance approach that any admissible reinsurance policy is dominated by a stop-loss reinsurance or a two-layer reinsurance, depending upon the amount of the reinsurance premium. Moreover, we obtain a similar result to Mossin's Theorem and find that it is optimal for the insurer to cede a loss as much as possible under the net premium principle. To further examine the reinsurance premium for the optimal piecewise linear reinsurance policy, we assume the expected value premium principle and derive the optimal reinsurance explicitly under (1) the criterion of minimizing the variance of the insurer's risk exposure, and (2) the criterion of minimizing the risk-adjusted value of the insurer's liability where the liability valuation is carried out using the cost-of-capital approach based on the conditional value at risk.

In this paper, we consider ℝd-valued integrable processes which are increasing in the convex order, i.e. ℝd-valued peacocks in our terminology. After the presentation of some examples, we show that an ℝd-valued process is a peacock if and only if it has the same one-dimensional marginals as an ℝd-valued martingale. This extends former results, obtained notably by Strassen [Ann. Math. Stat. 36 (1965) 423–439], Doob [J. Funct. Anal. 2 (1968) 207–225] and Kellerer [Math. Ann. 198 (1972) 99–122].

In this paper, we present a new proof of the celebrated theorem of Kellerer, stating that every integrable process, which increases in the convex order, has the same one-dimensional marginals as a martingale. Our proof proceeds by approximations, and calls upon martingales constructed as solutions of stochastic differential equations. It relies on a uniqueness result, due to Pierre, for a Fokker-Planck equation.

In this paper we further investigate the problem considered by Mizuno (2006) in the special case of identically distributed signals. Specifically, we first propose an alternative sufficient condition of crossing type for the convex order to hold between the conditional expectations given signal. Then, we prove that the bivariate (2,1)-increasing convex order ensures that the conditional expectations are ordered in the convex sense. Finally, the L2 distance between the quantity of interest and its conditional expectation given signal (or expected conditional variance) is shown to decrease when the strength of the dependence increases (as measured by the (2,1)-increasing convex order).

Let X1, X2,… and Y1, Y2,… be two sequences of absolutely continuous, independent and identically distributed (i.i.d.) random variables with equal means E(Xi)=E(Yi), i=1,2,… In this work we provide upper bounds for the total variation and Kolmogorov distances between the distributions of the partial sums ∑i=1nXi and ∑i=1nYi. In the case where the distributions of the Xis and the Yis are compared with respect to the convex order, the proposed upper bounds are further refined. Finally, in order to illustrate the applicability of the results presented, we consider specific examples concerning gamma and normal approximations.

Let S1 and S2 be two signals of a random variable X, where G1(s1 ∣ x) and G2(s2 ∣ x) are their conditional distributions given X = x. If, for all s1 and s2, G1(s1 ∣ x) - G2(s2 ∣ x) changes sign at most once from negative to positive as x increases, then the conditional expectation of X given S1 is greater than the conditional expectation of X given S2 in the convex order, provided that both conditional expectations are increasing. The stochastic order of the sufficient condition is equivalent to the more stochastically increasing order when S1 and S2 have the same marginal distribution and, when S1 and S2 are sums of X and independent noises, it is equivalent to the dispersive order of the noises.

Under certain assumptions on the dependence structure of the residual lives of the insured (i.e. their independence, positive association, or negative association), in this paper we establish some laws of large numbers for the convex upper bounds, derived by the technique of comonotonicity, of the present value function of a homogeneous portfolio composed of the whole-life insurance policies.

We characterize the (continuous) majorization of integrable functions introduced by Hardy, Littlewood, and Pólya in terms of the (discrete) majorization of finite-dimensional vectors, introduced by the same authors. The most interesting version of this result is the characterization of the (increasing) convex order for integrable random variables in terms of majorization of vectors of expected order statistics. Such a result includes, as particular cases, previous results by Barlow and Proschan and by Alzaid and Proschan, and, in a sense, completes the picture of known results on order statistics. Applications to other stochastic orders are also briefly considered.

We derive conditions under which the increments of a vector process are associated — i.e. under which all pairs of increasing functions of the increments are positively correlated. The process itself is associated if it is generated by a family of associated and monotone kernels. We show that the increments are associated if the kernels are associated and, in a suitable sense, convex. In the Markov case, we note a connection between associated increments and temporal stochastic convexity.

Our analysis is motivated by a question in variance reduction: assuming that a normalized process and its normalized compensator converge to the same value, which is the better estimator of that limit? Under some additional hypotheses we show that, for processes with conditionally associated increments, the compensator has smaller asymptotic variance.

A direct proof is presented for the fact that the stationary system queue length distribution just after the service completion epochs in the Mx/GI/1/k queue is given by the truncation of a measure on Z+ = {0, 1, ·· ·}. The related truncation formulas are well known for the case of the traffic intensity ρ < 1 and for the virtual waiting time process in M/GI/1 with a limited waiting time (Cohen (1982) and Takács (1974)). By the duality of GI/MY/1/k to Mx/GI/1/k + 1, we get a similar result for the system queue length distribution just before the arrival of a customer in GI/MY/1/k. We apply those results to prove that the loss probabilities of Mx/GI/1/k and GI/MY/1/k are increasing for the convex order of the service time and interarrival time distributions, respectively, if their means are fixed.