Hess C., Seri R., Choirat C. (2010), Ergodic theorems for extended real-valued random variables, Stochastic Processes and their Applications, 120, 10, p. 1908-1919
We first establish a general version of the Birkhoff Ergodic Theorem for quasi-integrable extended real-valued random variables without assuming ergodicity. The key argument involves the Poincaré Recurrence Theorem. Our extension of the Birkhoff Ergodic Theorem is also shown to hold for asymptotic mean stationary sequences. This is formulated in terms of necessary and sufficient conditions. In particular, we examine the case where the probability space is endowed with a metric and we discuss the validity of the Birkhoff Ergodic Theorem for continuous random variables. The interest of our results is illustrated by an application to the convergence of statistical transforms, such as the moment generating function or the characteristic function, to their theoretical counterparts.
Hess C. (2009), Computing the mean and the variance of the cedent's share for largest claims reinsurance covers, Insurance: Mathematics and Economics, 44, 3, p. 497-504
We present mathematical results allowing one to evaluate the moments of order 1 and 2 of the cedent's share in the framework of reinsurance treaties based on ordered claimsizes. These results consist of closed analytical formulas that do not involve any approximation procedure. This is illustrated by numerical examples when the claim number has the Poisson or the negative binomial distribution, and the claim cost has the exponential or the Pareto distribution.
Hess C. (2007), On the Parametrized Integral of a Multifunction: The Unbounded Case, Set-Valued Analysis, 15, 1, p. 1-20
Integration of set-valued maps (alias multifunctions) depending on a parameter is revisited. Results of Artstein, and of Saint-Pierre and Sajid are extended to the case of set-valued maps whose values may be unbounded. In the general case, this is achieved assuming that the transition probabilities involved in the integration procedure are absolutely continuous with respect to some fixed probability measure. However, when the integrating probability measure does not depend on the parameter this hypothesis is shown to be unnecessary. On the other hand, an alternative proof of a result of Saint-Pierre and Sajid is provided for convex compact-valued multifunctions. An application is given to the control of chattering systems. It is an extension of a result of Artstein to the case of set-valued maps with unbounded values. The proof of the main results is simple and essentially relies on measurable selections arguments.
Hess C., Couvreux J. (1999), A Lévy Type Martingale Convergence Theorem for Random Sets with Unbounded Values, Journal of Theoretical Probability, 12, 4, p. 933-969
Given a nondecreasing sequence (bernou n ) of sub-sgr-fields and a real or vector valued random variable f, the Lévy Martingale convergence Theorem (LMCT) asserts that E(f/bernou n ) converges to E(f/bernouinfin) almost surely and in L 1, where bernouinfin stands for the sgr-field generated by the bernou n . In the present paper, we study the validity of the multivalued analog this theorem for a random set F whose values are members of Fscr(X), the space of nonempty closed sets of a Banach space X, when Fscr(X) is endowed either with the Painlevé-Kuratowski convergence or its infinite dimensional extensions. We deduce epi-convergence results for integrands via the epigraphical multifunctions. As it is known, these results are useful for approximating optimization problems. The method relies on countability supportness hypotheses which are shown to hold when the values of the random set E(F/bernou n ) do not contain any line. On the other hand, since the values of F are not assumed to be bounded, conditions involving barrier and asymptotic cones are shown to be necessary. Moreover, we discuss the relations with other multivalued martingale convergence theorems and provide examples showing the role of the hypotheses. Even in the finite dimensional setting, our results are new or subsume already existing ones.
Hess C., Barbati A. (1998), The Largest Class of Closed Convex Valued Multifunctions for which Effros Measurability and Scalar Measurability Coincide, Set-Valued Analysis, 6, 3, p. 209-236
This paper deals with the comparison of Effros measurability and scalar measurability for multifunctions whose values lie in C(X), the set of closed convex subsets of a normed linear space X. An introductory counter-example shows that, on C(X), the Effros measurability is strictly stronger than the scalar measurability. Then, we introduce the notion of countably supported subspace of C(X). After some preparatory results and examples about this class of convex subsets, we show that on an analytic countably supported subspace of C(X), the Effros and the scalar C of C(X), nonnecessarily analytic, the Effros and scalar C is countably supported. This leads us to exhibit and study a wide class of subspaces of C(X) both countably supported and analytic. At last, we compare our results with the already existing ones and we briefly show how our main results can be extended to the case where X is a locally convex vector space.
Hess C. (1994), Multivalued strong laws of large numbers in the slice topology. Application to integrands, Set-Valued Analysis, 2, 1-2, p. 183-205
Starting from the multivalued strong law of large numbers in the Wijsman topology recently proved by the present author, we deduce two multivalued strong laws of large numbers in the 'slice topology' introduced by Beer. An application to integrands via their epigraphical multifunctions is also provided.
Hess C. (1991), On multivalued martingales whose values may be unbounded: martingale selectors and mosco convergence, Journal of Multivariate Analysis, 39, 1, p. 175-201
Using classical results on the projective limit of a sequence of subsets, we show the existence of martingale selectors for a multivalued martingale (and supermartingale) with closed values in a separable Banach space X. The existence of L1(X)-bounded or uniformly integrable martingale selectors is also discussed. At last, applications to the Mosco convergence of multivalued supermartingales and supermartingale integrands are provided.
Hess C. (1990), Measurability and integrability of the weak upper limit of a sequence of multifunctions, Journal of Mathematical Analysis and Applications, 153, 1, p. 226-249
We provide several properties of the weak upper limit of a sequence of subsets of a separable Banach space, such as a criterion of non-vacuity, of closedness, etc. We also examine the measurability of the multifunction defined as the weak upper limit of a sequence of multifunctions. At last, applications to the existence of a measurable and Bochner integrable selector for this multifunction are presented.
Saadoune M., Hess C., Castaing C. (2006), Tightness conditions and integrability of the sequential weak upper limit of a sequence of multifunctions, in Yamazaki A., Kusuoka S. (eds), Advances in mathematical economics, Berlin, Springer, p. 11-44
Various notions of tightness for measurable multifunctions are introduced and compared. They are used to derive results on the existence of integrable selections for the sequential weak upper limit of a sequence of multifunctions. Similar questions are examined for multifunctions with values in a dual space. Some results are particularized in the single-valued case, and applications to the multidimensional Fatou Lemma, both in the primal and in the dual space, are derived. This is achieved under conditions weaker than or noncomparable to L 1-boundedness.
Deléglise M-P., Hess C., Nouet S. (2009), Tarification, provisionnement et pilotage d'un contrat dépendance, Cahier de la Chaire "Risques et Chances de la Transition Démographique", Paris, Université Paris-Dauphine, 24
A partir des résultats de l'enquête "Handicap-Incapacités-Dépendance" du Ministère de la Santé et de scénarios d'évolution future de la dépendance des personnes âgées, on examine de manière prospective la tarification, le provisionnement et l'équilibre des comptes relatifs à un portefeuille de contrats dépendance. Les méthodes présentées et les résultats ont pour but d'aider l'assureur à piloter le portefeuille en fonction de l'évolution de l'environnement démographique et financier.
Based on the report "Handicap-Incapacités-Dépendance" of the French "Ministère de la Santé" and on scenarios for the disability of elder persons, the rating, the reserving and the financial balance relative to such a disability porfolio are examined in a prospective approach. The methods and the results aim at helping the insurer to control the portfolio according to the evolution of the demographic and financial environment.