Barbara Martinucci

Compound Poisson process with a Poisson subordinator

A compound Poisson process whose randomized time is an independent Poisson process is called compound Poisson process with Poisson subordinator. We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps and normal jumps. Then for the iterated Poisson process we discuss some properties and provide convergence results to a Poisson process. The first-crossing-time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasing and constant boundary, where we provide some closed-form results, and (ii) a linearly increasing boundary, where we propose an iterative procedure to compute the first-crossing-time density and survival functions.

On the damped geometric telegrapher’s process

The geometric telegrapher’s process has been proposed in 2002 as a model to describe the dynamics of the price of risky assets. In this contribution we consider a related stochastic process, whose trajectories have two alternating slopes, for which the random times between consecutive slope changes have exponential distribution with linearly increasing parameters. This leads to a process characterized by a damped behavior. We study the main features of the transient probability law of the process, and of its stationary limit.

On a Symmetric, Nonlinear Birth-Death Process with Bimodal Transition Probabilities

We consider a bilateral birth-death process having sigmoidal-type rates. A thorough discussion on its transient behaviour is given, which includes studying symmetry properties of the transition probabilities, finding conditions leading to their bimodality, determining mean and variance of the process, and analyzing absorption problems in the presence of 1 or 2 boundaries. In particular, thanks to the symmetry properties we obtain the avoiding transition probabilities in the presence of a pair of absorbing boundaries, expressed as a series.

A FIRST-PASSAGE-TIME PROBLEM FOR SYMMETRIC AND SIMILAR TWO-DIMENSIONAL BIRTH-DEATH PROCESSES

A spatial symmetry property of a two-dimensional birth–death process X(t) with constant rates is exploited in order to obtain closed-form expressions for first-passage-time densities through straight-lines x 2 = x 1 + r and for the related taboo transition probabilities. An analogous study is performed on a birth–death process with state-dependent rates that is similar to X(t) in the sense that the ratio of their transition functions is time independent. Examples of applications to double-ended queues and stochastic neuronal modeling are also provided.

ON A FIRST-PASSAGE-TIME PROBLEM FOR THE COMPOUND POWER-LAW PROCESS

We consider a first-passage-time problem for a compound Poisson process characterized by independent, identically and exponentially distributed jumps, occurring according to the power-law process (PLP). First of all, we refer to the conditional product moments of arrival times and to the interarrival times density of a power-law process. We then obtain the probability density of the crossing time through a linear boundary at the occurrence of the nth jump. In particular, we express the first-passage-time density in terms of a conditional expectation involving the arrival times.

Fractional Growth Process with Two Kinds of Jumps

We consider a suitable fractional jump process describing growth phenomena, that may be viewed as a counting process characterized by 2 kinds of jumps with size 1 and 2. We obtain the probability generating function and the probability law of the process, expressed in terms of the generalized Mittag-Leffler function. The mean, variance, and squared coefficient of variation are also provided.

On the Geometric Brownian Motion with Alternating Trend

A basic model in mathematical finance theory is the celebrated geometric Brownian motion. Moreover, the geometric telegraph process is a simpler model to describe the alternating dynamics of the price of risky assets. In this note we consider a more general stochastic process that combines the characteristics of such two models. Precisely, we deal with a geometric Brownian motion with alternating trend. It is defined as the exponential of a standard Brownian motion whose drift alternates randomly between a positive and a negative value according to a generalized telegraph process. We express the probability law of this process as a suitable mixture of Gaussian densities, where the weighting measure is the probability law of the occupation time of the underlying telegraph process.

A generalized telegraph process with velocity driven by random trials

We consider a random trial-based telegraph process, which describes a motion on the real line with two constant velocities along opposite directions. At each epoch of the underlying counting process the new velocity is determined by the outcome of a random trial. Two schemes are taken into account: Bernoulli trials and classical Pólya urn trials. We investigate the probability law of the process and the mean of the velocity of the moving particle. We finally discuss two cases of interest: (i) the case of Bernoulli trials and intertimes having exponential distributions with linear rates (in which, interestingly, the process exhibits a logistic stationary density with nonzero mean), and (ii) the case of Pólya trials and intertimes having first gamma and then exponential distributions with constant rates.

Feedback effects in simulated stein's coupled neurons

A network consisting of two Stein-type neuronal units is analyzed under the assumption of a complete interaction between the neurons. The firing of each neuron causes a jump of constant amplitude of the membrane potential of the other neuron. The jump is positive or negative according to whether the firing neuron is excitatory or inhibitory. Making use of a simulation procedure designed by ourselves, we study the interspike intervals of the two neurons by means of their histograms, of some descriptive statistics and of empirical distribution functions. Furthermore, via the crosscorrelation function, we investigate the synchronization between the neurons firing activity in the special case when one neuron is excitatory and the other is inhibitory.

A Neuronal Model with Excitatory and Inhibitory Inputs Governed by a Birth-Death Process

A stochastic model for the firing activity of a neuronal unit has been recently proposed in [4]. It includes the decay effect of the membrane potential in the absence of stimuli, and the occurrence of excitatory inputs driven by a Poisson process. In order to add the effects of inhibitory stimuli, we now propose a Stein-type model based on a suitable exponential transformation of a bilateral birth-death process on