Università di Bologna

Dipartimento di Matematica

GRUPPO DI RICERCA DI PROBABILITÀ

“Processi Stocastici e applicazioni a Filtraggio,
Controllo, Simulazione e Finanza Matematica”

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The main interests of Group of Probability at the Mathemtics Department of Bologna are the theory of random fields and percolation and general problems on sequences of random variables also in connection with sequential sampling in statistics.

Members of the group

Prof. Massimo Campanino

professore ordinario

Dott. Irene Crimaldi

ricercatore

Dott. Serena Fuschini

assegnista

Research Topics

The theory of random fields can be considered a part or an extension of the geneal theory of stochastic processes. We are in particular interested in the study of Gibbs random fields and of percolation. Gibbs random fields are defined in terms of coherent systems of conditional probability distributions, also called specifications. We are also interested in general problems of probability on sequences of random variables also in connection with sequential sampling in statistics.

For what concerns random fields the research related to Ornstein-Zernike theory, that was rigorously developed for percolation in the work of M. Campanino and D. Ioffe in the work "Ornstein-Zernike Theory for the Bernoulli Bond Percolation on Z^d" ([CI]) that appeared in The Annals of Probability, was extended to the cases of Ising random fields with finite range potential in the work by M. Campanino, D. Ioffe and Y. Velenik "Ornstein-Zernike Theory for finite range Ising models above Tc" ([CIV1]) that appeared on Probability Theory and Related Fields. In the work "Fluctuation Theory of Connectivities for Subcritical Random Cluster Models" by M. Campanino, D. Ioffe and Y. Velenik, that appears in The Annals of Probability, a theory of fluctuations is developed for models of random clusters below the critical pointdom clusters below the critical point. The theory is based on a probabilistic description of long connected clusters in terms of essentially one-dimensional chains of irreducible objects. Statistics of local observables, such as displacement, on these chains obey classical limit laws and this construction leads to a representation of percolation clusters in terms of random walks. Other results are obtained in the two-dimensional case by using duality. In collaboration with Dr. Michele Gianfelice of the University of Calabria in the framework of Bernoulli percolation we have studied the probability that three sites belong to the same percolation cluster in the limit as their distances tend to infinity. Moreover we have determined the typical realizations of this event also in the limit as the distances of the points tend to infinity. This research is based on the results of the work [CI] on the asymptotic behaviour of the connection probability of two points and on the techniques and constructions that were introduced there. The result will appear on Theory of Probability and Related Fields.

Dr. Irene Crimaldi in a series of works ([Cr], [CrP], [Cr2], [CrP2], [CrP3] [CrP4], [CrLP1]) studied the notion of convergence of conditional explectations and its applications to Filtering Theory and to the Theory of Stochastic Processes. Recently she is studying urn models ([BACr])and sequences of species sampling. Moreover, in collaboration with G. Letta and L. Pratelli([CrLP2]), she studied a problem on convergence of sequences of sigma fields that in the past has studied by other mathematicians, among which recently by Yor, with application to the theorem of convergence of inverse martingales.

For what concerns the study od stochastic processes with financial applications, in a joint work M. Campanino, F. Biagini (presently at Ludwig Maximillian University of Munich) and S. Fuschini in paper [BCF] introduce a discrete approximation for the stochastic integral with respect to the fractional Brownian motion of Hurst index H>1/2 defined in terms of the divergence operator. In order to determine a suitable class of integrands for which the approximation holds, the relations between the spaces of Malliavin differentiable processes in the fractional and standard case are also investigated in the paper. Dr. Serena Fuschini, in collaboration with prof. Francesca Biagini, is now studying models of the risk of credit based on the Fractional Brownian motion.