Amelia Giuseppina Nobile

A NEW INTEGRAL EQUATION FOR THE EVALUATION OF FIRST-PASSAGE-TIME PROBABILITY DENSITIES

The first-passage-time p.d.f. through a time-dependent boundary for one-dimensional diffusion processes is proved to satisfy a new Volterra integral equation of the second kind involving two arbitrary continuous functions. Use of this equation is made to prove that for the Wiener and the Ornstein-Uhlenbeck processes the singularity of the kernel can be removed by a suitable choice of these functions. A simple and efficient numerical procedure for the solution of the integral equation is provided and its convergence is briefly discussed. Use of this equation is finally made to obtain closed-form expressions for first-passage-time p.d.f.s in the case of various time-dependent boundaries.

On the M / M /1 Queue with Catastrophes and Its Continuous Approximation

For the M/M/1 queue in the presence of catastrophes the transition probabilities, densities of the busy period and of the catastrophe waiting time are determined. A heavy-traffic approximation to this discrete model is then derived. This is seen to be equivalent to a Wiener process subject to randomly occurring jumps for which some analytical results are obtained. The goodness of the approximation is discussed by comparing the closed-form solutions obtained for the continuous process with those obtained for the M/M/1 catastrophized queue.

On the evaluation of first-passage-time probability densities via non-singular integral equations

The algorithm given by Buonocore et al. (1) to evaluate first-passage-time p.d.f.s for Wiener and Ornstein-Uhlenbeck processes through a time-dependent boundary is extended to a wide class of time-homogeneous one-dimensional diffusion proc- esses. Several examples are thoroughly discussed along with some computational results. equation was proposed to determine first-passage-time p.d.f.s through time- dependent boundaries for time-homogeneous one-dimensional diffusion processes with natural boundaries. This equation was seen to be particularly manageable for computational purposes in the cases of Wiener and Ornstein-Uhlenbeck processes. Indeed, for these the kernel of the integral equation can be made continuous, thus overcoming some crucial difficulties arising from the singular nature of the integral equations for the first-passage-time p.d.f. Such continuity was made possible by a suitable choice of two arbitrary time-dependent functions that appear in the kernel of the integral equations. Here, in view of the relevance of first-passage-time problems in numerous applied fields (see, for instance, Buonocore et al. (1) and references therein), we intend to provide an extension of the results of Buonocore et al. (1) in two directions: on the one hand, we shall prove that the method for regularizing the kernel of the integral equation is valid not only for the Wiener and the Ornstein-Uhlenbeck processes but can also be used for a more general class of diffusion processes whose free transition

ON THE ASYMPTOTIC BEHAVIOUR OF FIRST- PASSAGE-TIME DENSITIES FOR ONE-DIMENSIONAL DIFFUSION PROCESSES AND VARYING BOUNDARIES

Making use of the integral equations given in [1], [2] and [3], the asymptotic behaviour of the first-passage time (FPT) p.d.f.s through certain time-varying boundaries, including periodic boundaries, is determined for a class of onedimensional diffusion processes with steady-state density. Sufficient conditions are given for the cases both of single and of pairs of asymptotically constant and asymptotically periodic boundaries, under which the FPT densities asymptotically exhibit an exponential behaviour. Explicit expressions are then worked out for the processes that can be obtained from the Omstein-Uhlenbeck process by spatial transformations. Some new asymptotic results for the FPT density of the Wiener process are finally proved, together with a few miscellaneous results.

Exponential trends of first passage time densities for a class of diffusion processes with steady-state distribution

The asymptotic behavior of the first-passage-time p.d.f. through a constant boundary is studied when the boundary approaches the endpoints of the diffusion interval. We show that for a class of diffusion processes possessing a steady-state distribution this p.d.f. is approximately exponential, the mean being the average first-passage time to the boundary. The proof is based on suitable recursive expressions for the moments of the first-passage time. PASSAGE-TIME MOMENTS

ON THE TWO-BOUNDARY FIRST-CROSSING-TIME PROBLEM FOR DIFFUSION PROCESSES

The first-crossing-time problem through two time-dependent boundaries for one-dimensional diffusion processes is considered. The first-crossing p.d.f.s from below and from above are proved to satisfy a new system of Volterra integral equations of the second kind involving two arbitrary continuous functions. By a suitable choice of such functions a system of continuous-kernel integral equations is obtained and an efficient algorithm for its solution is provided. Finally, conditions on the drift and infinitesimal variance of the diffusion process are given such that the system of integral equations reduces to a non-singular single integral equation for the first-crossing-time p.d.f.

A Double-ended Queue with Catastrophes and Repairs, and a Jump-diffusion Approximation

Consider a system performing a continuous-time random walk on the integers, subject to catastrophes occurring at constant rate, and followed by exponentially-distributed repair times. After any repair the system starts anew from state zero. We study both the transient and steady-state probability laws of the stochastic process that describes the state of the system. We then derive a heavy-traffic approximation to the model that yields a jump-diffusion process. The latter is equivalent to a Wiener process subject to randomly occurring jumps, whose probability law is obtained. The goodness of the approximation is finally discussed.

On a symmetry-based constructive approach to probability densities for two-dimensional diffusion processes

The method earlier introduced for one-dimensional diffusion processes [6] is extended to obtain closed form expressions for the transition p.d.f.s of two-dimensional diffusion processes in the presence of absorbing boundaries and for the first-crossing time p.d.f.s through such boundaries. Use of such a method is finally made to analyse a two-dimensional linear process.

Diffusion approximation to a queueing system with time-dependent arrival and service rates

A time nonhomogeneous diffusion approximation to a single server-single queue service system is obtained. Under various assumptions on the time-dependent functions appearing in the infinitesimal moments, transient and steady-state behaviour are analyzed. In particular, a diffusion approximation characterized by space-linear and time-varying moments is studied. The density of the busy period and the probability for the busy period to terminate are obtained. Finally, estimates of the goodness of the diffusion approximation are given.

On Gompertz growth model and related difference equations

Within the context of the dynamics of populations described by first order difference equations a datailed study of the Gompertz growth model is performed. This is mainly achieved by proving several theorems for a class of difference equations generalizing the Gompertz equation. Some interesting features of the discrete Gompertz model, not exhibited by other well known growth models, are finally pointed out.