Time-Inhomogeneous Feller-type Diffusion Process with Absorbing Boundary Condition

AbstractA time-inhomogeneous Feller-type diffusion process with linear infinitesimal drift $$\alpha (t)x+\beta (t)$$ α ( t ) x + β ( t ) and linear infinitesimal variance 2r(t)x is considered. For this process, the transition density in the presence of an absorbing boundary in the zero-state and the first-passage time density through the zero-state are obtained. Special attention is dedicated to the proportional case, in which the immigration intensity function $$\beta (t)$$ β ( t ) and the noise intensity function r(t) are connected via the relation $$\beta (t)=\xi \,r(t)$$ β ( t ) = ξ r ( t ) , with $$0\le \xi <1$$ 0 ≤ ξ < 1 . Various numerical computations are performed to illustrate the effect of the parameters on the first-passage time density, by assuming that $$\alpha (t)$$ α ( t ) , $$\beta (t)$$ β ( t ) or both of these functions exhibit some kind of periodicity

Exact solutions and asymptotic behaviors for the reflected Wiener, Ornstein-Uhlenbeck and Feller diffusion processes

We analyze the transition probability density functions in the presence of a zero-flux condition in the zero-state and their asymptotic behaviors for the Wiener, Ornstein Uhlenbeck and Feller diffusion processes. Particular attention is paid to the time-inhomogeneous proportional cases and to the time-homogeneous cases. A detailed study of the moments of first-passage time and of their asymptotic behaviors is carried out for the time-homogeneous cases. Some relationships between the transition probability density functions for the restricted Wiener, Ornstein-Uhlenbeck and Feller processes are proved. Specific applications of the results to queueing systems are provided

Inference on a stochastic two-compartment model in tumor growth

A continuous-time model that incorporates several key elements in tumor dynamics is analyzed. More precisely, the form of proliferating and quiescent cell lines comes out from their relations with the whole tumor mass, giving rise to a two-dimensional diffusion process, generally time non-homogeneous. This model is able to include the effects of the mutual interactions between the two subpopulations. Estimation of the rates of the two subpopulations based on some characteristics of the involved diffusion processes is discussed when longitudinal data are available. To this aim, two procedures are presented. Some simulation results are developed in order to show the validity of these procedures as well as to compare them. An application to real data is finally presented

A new approach to the construction of first-passage-time densities

A new method for constructing first-passage-time probability density functions is outlined. This rests on the possibility of constructing the transition p.d.f. of a new diffusion process in terms of a preassigned transition p.d.f. without making use of the classical space-time transformations of the Kolmogorov equation. A few examples are finally discusse

Study of a general growth model

We discuss a general growth curve including several parameters, whose choice leads to a variety of models including the classical cases of Malthusian, Richards, Gompertz, Logistic and some their generalizations. The advantage is to obtain a single mathematically tractable equation from which the main characteristics of the considered curves can be deduced. We focus on the effects of the involved parameters through both analytical results and computational evaluations

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.Comment: 18 pages, 5 figures, paper accepted by "Methodology and Computing in Applied Probability", the final publication is available at http://www.springerlink.co

Time-Inhomogeneous Feller-Type Diffusion Process in Population Dynamics

The time-inhomogeneous Feller-type diffusion process, having infinitesimal drift α(t)x+β(t) and infinitesimal variance 2r(t)x, with a zero-flux condition in the zero-state, is considered. This process is obtained as a continuous approximation of a birth-death process with immigration. The transition probability density function and the related conditional moments, with their asymptotic behaviors, are determined. Special attention is paid to the cases in which the intensity functions α(t), β(t), r(t) exhibit some kind of periodicity due to seasonal immigration, regular environmental cycles or random fluctuations. Various numerical computations are performed to illustrate the role played by the periodic functions

On the Absorbing Problems for Wiener, Ornstein&ndash;Uhlenbeck, and Feller Diffusion Processes: Similarities and Differences

For the Wiener, Ornstein&ndash;Uhlenbeck, and Feller processes, we study the transition probability density functions with an absorbing boundary in the zero state. Particular attention is paid to the proportional cases and to the time-homogeneous cases, by obtaining the first-passage time densities through the zero state. A detailed study of the asymptotic average of local time in the presence of an absorbing boundary is carried out for the time-homogeneous cases. Some relationships between the transition probability density functions in the presence of an absorbing boundary in the zero state and between the first-passage time densities through zero for Wiener, Ornstein&ndash;Uhlenbeck, and Feller processes are proven. Moreover, some asymptotic results between the first-passage time densities through zero state are derived. Various numerical computations are performed to illustrate the role played by parameters

A Stochastic Gompertz Model with Jumps for an Intermittent Treatment in Cancer Growth

To analyze the effect of a therapeutic program that provides intermittent suppression of cancer cells, we suppose that the Gompertz stochastic diffusion process is influenced by jumps that occur according to a probability distribution, producing instantaneous changes of the system state. In this context a jump represents an application of the therapy that leads the cancer mass to a return state randomly chosen. In particular, constant and exponential intermittence distribution are considered for different choices of the return state. We perform several numerical analyses to understand the behavior of the process for different choices of intermittence and return point distributions