Bruno Girotto

Chebyshev type inequality for Choquet integral and comonotonicity

We supply a Chebyshev type inequality for Choquet integral and link this inequality with comonotonicity.

A Chebyshev type inequality for Sugeno integral and comonotonicity

We supply a characterization of comonotonicity property by a Chebyshev type inequality for Sugeno integral.

Numeraires, Equivalent Martingale Measures and Completeness in Finite Dimensional Securities Markets

Abstract A numeraire is a portfolio that, when securities, prices and dividends are expressed in its units, admits an equivalent martingale measure transforming any gain process into a martingale. We show that the set of equivalent martingale measures of a numeraire is one-to-one with a subset of Arrow-Debreu state prices, which becomes the whole set if and only if the numeraire is self-financing. Hence our result extends those (e.g. Harrison and Kreps ( Journal of Economic Theory , 1979, 20, 381–408) Dothan ( Prices in Financial Markets , Oxford Univ. Press, New York, 1990)) stated for specific self-financing numeraires. We also identify markets admitting self-financing numeraires, and characterize completeness, in terms of equivalent martingale measures, without requiring that specific securities be traded.

A characterization of neo-additive measures

Neo-additive and generalized neo-additive capacities were introduced in order to capture both optimistic and pessimistic attitudes towards uncertainty without abandoning the subjective probabilistic approach. In this way, one can obtain, as particular cases, some well-known decision criteria (via Choquet expectation) adopted in Decision Theory and Mathematical Statistics.In order to introduce these capacities, Chateauneuf, Eichberger, Grant and Eichberger, Grant, Lefort consider three types of events: universal, null and essential events; afterwards they introduce capacities which are null on null events (null property), assume value one on universal events (normalization property) and are translations of finitely additive probabilities on the family of essential events. Finally, they supply a theoretic measure characterization of these type of capacities.In this paper, we introduce neo-additive measures as monotone measures which are translations of finitely additive ones on the family of essential events, without assumption of normalization property and null property. Moreover, we supply a simple and natural theoretic characterization of these measures obtaining, as particular cases, the corresponding results of the previous authors. In this way, our results give a robust foundation of neo-additive and generalized neo-additive capacities in abstract measure setting.

Generic Existence and Robust Nonexistence of Numeraires in Finite Dimensional Securities Markets

A numeraire is a portfolio that, if prices and dividends are denominated in its units, admits an equivalent martingale measure that transforms all gains processes into martingales. We first supply a necessary and sufficient condition for the generic existence of numeraires in a finite dimensional setting. We then characterize the arbitrage-free prices and dividends for which the absence of numeraires survives any small perturbation preserving no arbitrage. Finally, we identify the cases when any small, but otherwise arbitrary, perturbation of prices and dividends preserves either the existence of numeraires, or their nonexistence under no arbitrage.

Regular and purely irregular bounded charges: a decomposition theorem

We introduce the notions of regular and purely irregular charges with respect to a pair of pavings and study their structural properties. Moreover, we link regularity and σ-additivity, obtaining some generalizations of well-known theorems. Finally, when the pavings satisfy some reasonable weak conditions, we can decompose any bounded charge into regular and purely irregular decomposants; this decomposition becomes the Hewitt-Yosida one, whenever the charges are defined on the Baire σ-field of a countably compact space