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