Seminar Details
2007-04-19 (16.30pm) : Colloquium CESAME/MAPA "Integral stochastic orderings for comparing actuarial risks and dependence structures."
At Auditoire Ch.-J de La Vallée Poussin (cycl 01)
Organized by Mathematical Engineering
Speaker :
Michel Denuit (UCL)
Abstract :
This lecture aims to discuss various stochastic inequalities and their applications in risk management.
Broadly speaking, a stochastic ordering is an order relation that allows to compare random variables.
Integral stochastic orderings are defined by reference to classes of measurable functions and inherit
their properties from the defining class. According to the class of functions under consideration, integral
stochastic orderings aim to express intuitive ideas like "being smaller than" or "being less variable than", for instance.
Integral stochastic orderings can also be used to compare random vectors. In this case, it is often convenient to distinguish
the marginal behavior from the underlying dependence structure (described by the copula function in the continuous case).
Dependence orders allow to compare the strength of dependence and to express the fact that the components of some random
vector are « more positively dependent » than those of another random vector. In the management of insurance portfolios, the main
risk is the simultaneous occurrence of large losses. This behavior is appropriately described by various notions of positive dependence,
like co-monotonicity, multivariate total positivity, conditional increasingness, association, positive quadrant dependence, etc. Such dependence
notions are closely related to dependence orders. Several applications of stochastic orders are discussed.