E. Di Nardo

Umbral nature of the Poisson random variables

The symbolic method, nowadays known as umbral calculus, has been extensively used since the nineteenth century although the mathematical community was sceptical of it, perhaps because of its lack of foundation. This method was fully developed by Rev. John Blissard in a series of papers beginning in 1861 [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]; nevertheless it is impossible to attribute the credit for the original idea to him alone since Blissard’s calculus has a mathematical source in symbolic differentiation. In [22] Lucas even claimed that the umbral calculus has its historical roots in the writing of Leibniz for the successive derivatives of a product with two or more factors; moreover Lucas held that this symbolic method had been developed subsequently by Laplace, by Vandermonde, by Herschel, and augmented by the works of Cayley and of Sylvester in the theory of forms. Lucas’ papers attracted considerable attention and the predominant contribution of Blissard to this method was kept in the background. Bell reviewed the whole subject in several papers, restoring the purport of the Blissard’s idea [4], and in 1940 he tried to give a rigorous foundation for the mystery at the bottom of the umbral calculus [5] but his attempt did not gain a hold. Indeed, in the first modern textbook of combinatorics [24], Riordan largely employed this symbolic method without giving any formal justification. It was Gian-Carlo Rota [26] who six years later disclosed the “umbral magic art” consisting in lowering and raising exponents bringing to the light the underlying linear functional. The ideas from [26] led Rota and his collaborators to conceive a beautiful theory which gave rise to a large variety of applications ([23, 27]). Some years later, Roman and Rota [25] gave rigorous form to the umbral tricks in the setting of Hopf algebras. On the other hand, as Rota himself has written in [28]: “…Although the notation of Hopf algebra satisfied the most ardent advocate of spic-and-span rigor, the translation of “classical”, umbral calculus into the newly found rigorous language made the method altogether unwieldy and unmanageable. Not only was the eerie feeling of witchcraft lost in the translation, but, after such a translation, the use of calculus to simplify computation and sharpen our intuition was lost by the wayside…” Thus in 1994 Rota and Taylor [28] started a rigorous and simple presentation of the umbral calculus in the spirit of the founders. The present article takes this last point of view.

On some computational results for single neurons' activity modeling.

The classical Ornstein-Uhlenbeck diffusion neuronal model is generalized by inclusion of a time-dependent input whose strength exponentially decreases in time. The behavior of the membrane potential is consequently seen to be modeled by a process whose mean and covariance classify, it as Gaussian-Markov. The effect of the input on the neurons firing characteristics is investigated by comparing the firing probability densities and distributions for such a process with the corresponding ones of the Ornstein-Uhlenbeck model. All numerical results are obtained by implementation of a recently developed computational method.

Simulation of First-Passage Times for Alternating Brownian Motions

The first-passage-time problem for a Brownian motion with alternating infinitesimal moments through a constant boundary is considered under the assumption that the time intervals between consecutive changes of these moments are described by an alternating renewal process. Bounds to the first-passage-time density and distribution function are obtained, and a simulation procedure to estimate first-passage-time densities is constructed. Examples of applications to problems in environmental sciences and mathematical finance are also provided.

On the Asymptotic Behavior of First Passage Time Densities for Stationary Gaussian Processes and Varying Boundaries

Making use of a Rice-like series expansion, for a class of stationary Gaussian processes the asymptotic behavior of the first passage time probability density function through certain time-varying boundaries, including periodic boundaries, is determined. Sufficient conditions are then given such that the density asymptotically exhibits an exponential behavior when the boundary is either asymptotically constant or asymptotically periodic.

Symbolic solutions of some linear recurrences

Abstract A symbolic method for solving linear recurrences of combinatorial and statistical interest is introduced. This method essentially relies on a representation of polynomial sequences as moments of a symbol that looks as the framework of a random variable with no reference to any probability space. We give several examples of applications and state an explicit form for the class of linear recurrences involving Sheffer sequences satisfying a special initial condition. The results here presented can be easily implemented in a symbolic software.

On a Symbolic Version of Multivariate Lévy Processes

By using the classical umbral calculus, we propose a symbolic expression of multivariate Levy processes. Compared with the classical one, the advantage of this approach relies in relaxing many of usual hypothesis necessary to deal with stochastic processes. As example, we recover the symbolic representation of multivariate Brownian motion and multivariate compound Poisson process. Open problems are also addressed.

Evaluation of neuronal firing densities via simulation of a jump-diffusion process

We consider a stochastic neuronal model in which the time evolution of the membrane potential is described by a Wiener process perturbed by random jumps driven by a counting process. We consider the first-crossing-time problem through a constant boundary for such a process, in order to describe the firing activity of the model neuron. We build up a new simulation procedure for the construction of firing densities estimates.