Stochastic Models for Complex Systems

The research targets will be achieved through a collective effort among the scholars of the three units. In relationship with the previously described project targets, Napoli research unit will mainly develop the following specific sub-tasks:

WP1: The univariate distorted distribution is useful in risk theory to represent changes (distortions) in the expected distributions of some risks. They are also applied to represent distributions of order statistics, coherent systems, proportional hazard rate and proportional reversed hazard rate models. Our study is to extend this concept to the multivariate setup, because they are a valid alternative to the copula representations especially when the marginal distributions have complicated expressions. Our aim is to study paired (dependent) ordered data, joint residual lifetimes, order statistics and coherent systems, in collaboration with Salento Unit.

WP2: In the context of reliability theory, we propose to study and analyse new measures of information taking into account the expectation of the information content and its variability. We focus our attention on several versions of the entropy, or Shannon entropy, and its dual measure, i.e. the extropy, such as their weighted and cumulative versions. We use these measures of information to handle the uncertainty also about the past lifetime, the inactivity time and the residual lifetime. In a more general context like the Dempster-Shafer theory of evidence, characterized by more uncertainty, we introduce measures of information based on basic probability assignment. We also focus our study on other reliability properties, such as aging properties, involving the hazard rate and the reversed hazard rate functions. In the study of coherent systems, policy plans are investigating for the warranty length of a repairable system which consists of a number of minimal repairs.

WP3: In the context of stochastic models to describe growth of populations using tools of Fractional Calculus, we designed stochastic versions for some fractional Gompertz curves in order to include memory effects in the classical models. We studied a class of linear fractional-integral stochastic equations involved in such dynamics. These equations are used to construct fractional stochastic Gompertz models. A new fractional Gompertz model is introduced and a stochastic version of it is provided. Analysis and specializations of this are currently object of our investigations.

WP5: The distribution of the first exit time from an arbitrary open set was obtained for a class of semi-Markov processes as time-changed Markov processes. The asymptotic behavior of the survival function (for large t) and of the distribution function (for small t) was studied and some conditions for absolute continuity are provided. The use of semi-Markov models appears to be realistic under several aspects, for example, it makes the times between successive events a r.v. with infinite expectation, which is a desirable property. We are also studying some classes of continuous and discrete processes with the time changed by using subordinators in order to construct new classes of semi-Markov processes and to arrange their theoretical apparatus.

WP6: We adopted semi-Markov processes for neuronal modeling. Specifically, we used semi-Markov process (see WP5) obtained from time-changed Markov processes by using the inverse of a stable subordinator. We characterized fpt’s density of such process, deriving asymptotic behaviors such as the power-law decay of the related densities. We also focused on a fractional Ornstein-Uhlenbeck process involving a stochastic forcing term in the drift, as a solution of a linear stochastic differential equation driven by a fractional Brownian motion. For such process we specified mean and covariance functions, concentrating on their asymptotic behavior. Such a process showed a sort of short- or long-range dependence, under specified hypotheses on the covariance of the forcing process. Such properties reveal especially useful for neuronal models including some kind of memory and correlation effects. Further applications in neuronal modeling are currently investigated and more specialized fractional processes are under consideration. Numerical approximations and simulation algorithms of such processes and of their laws constitute alternative investigation approaches, but also validation tools.