Enrico Ferri, Carlos Vázquez, José Luis Fernández: Consistency and qualitative robustness, two of the main forms of stability usually required when dealing with risk estimators, are presented in an overall perspective by considering different notions of probabilistic symmetries at the level of the stochastic drivers. For such a purpose, a general topological framework is properly introduced by extending the common notion of the weak topology of measures. In particular, the main objective of the work is to state all the results in terms of exchangeability, the celebrated notion of distributional invariance arising when dealing with arbitrary permutations of finite order. In this respect, a new concept of qualitative robustness is defined in terms of the perturbations that affect the system when data are slightly perturbed, and a refined version of the celebrated Hampel’s theorem is provided. Such an overture turns out to be strongly appealing from the conceptual point of view, since data are the main drivers of the empirical analysis.

# ESR1 Enrico Ferri

Let E be a space of observables in a sequence of trials zeta_n and define m_n to be the empirical distributions of the outcomes. We discuss the almost sure convergence of the sequence m_n in terms of the psi -weak topology of measures, when the sequence zeta n is assumed to be stationary. In this respect, the limit variable is naturally described as a certain canonical conditional distribution. Then, given some functional tau defined on a space of laws, the consistency of the estimators tau(m_n) is investigated. Hence, a criterion for a refined notion of robustness, that applies when considering random measures, is provided in terms of the modulus of continuity of tau.

We consider the problem of seeking an optimal set of model points associated to a fixed portfolio of life insurance policies. Such an optimal set is characterized by minimizing a certain risk functional, which gauges the average discrepancy with the fixed portfolio in terms of the fluctuation of the interest rate term structure within a given time horizon. We prove a representation theorem which provides two alternative formulations of the risk functional and which may be understood in connection with the standard approaches for the portfolio immunization based on sensitivity analysis. For this purpose, a general framework concerning some techniques of stochastic integration in Banach space and Malliavin calculus is introduced. A numerical example is discussed when considering a portfolio of whole life policies.

José L. Fernández, Enrico Ferri, Carlos Vázquez: Let E be a space of observables in a sequence of trials Epsilon_n and define m_n to be the empirical distributions of the outcomes. We discuss the almost sure convergence of the sequence m_n in terms of the psi-weak topology of measures, when the sequence Epsilon_n is assumed to be stationary. In this respect, the limit variable is naturally described as a certain canonical conditional distribution. Then, given some functional tau defined on a space of laws, the consistency of the estimators tau(m_n) is investigated. Moreover, a result on the asymptotic stability of the distribution of tau(m_n) w.r.t. certain transformations of the data is given.

Ana M. Ferreiro, Enrico Ferri, José A. García-Rodríguez, Carlos Vázquez: This work deals with the automatic selection of model points portfolios of life insurance policies that reproduce the original portfolio, in the sense that they retain the market risk properties of the initial portfolio. In order to achieve this goal, we first propose a risk functional that incorporates the uncertain evolution of forward LIBOR rates to the portfolios of life insurance policies.