Mathematical and Statistical Methods for Actuarial Sciences and Finance

Some classes of multivariate risk measures

Marta Cardin and Elisa Pagani

Abstract.In actuarial literature the properties of risk measures or insurance premium prin-

ciples have been extensively studied. We propose a new kind of stop-loss transform and a

related order in the multivariate setting and some equivalent conditions. In our work there is

a characterisation of some particular classes of multivariate and bivariate risk measures and a

new representation result in a multivariate framework.

Key words:risk measures, distortion function, concordance measures, stochastic orders

1 Introduction

In actuarial sciences it is fairly common to compare two random variables that are

risks by stochastic orderings defined using inequalities on expectations of the random

variables transformed by measurable functions. By characterising the considered set

of functions some particular stochastic orderings may be obtained such as stochastic

dominance or stop-loss order. These stochastic order relations of integral form may

be extended to cover also the case of random vectors.

The main contribution of this paper concerns the construction of a mathematical

framework for the representation of some classes of multivariate risk measures; in

particular we study the extension to the multivariate case of distorted risk measures

and we propose a new kind of vector risk measure. Moreover, we introduce the product

stop-loss transform of a random vector to derive a multivariate product stop-loss order.

2 Multivariate case

We consider only non-negative random vectors. Let be the space of the states

of nature,Fbe theσ-field andPbe the probability measure onF. Our random

vector is the functionX: →Rn+such thatX(ω)represents the payoff obtained

if stateωoccurs. We also specify some notations:FX(x):Rn →[0,1] is the

distribution function ofX,SX(x):Rn→[0,1] is its survival or tail function, and

(X(ω)−a)+=max(X(ω)−a, 0 )componentwise.

M. Corazza et al. (eds.), Mathematical and Statistical Methodsfor Actuarial Sciencesand Finance
© Springer-Verlag Italia 2010