Simulation of sample paths for Gauss-Markov processes in the presence of a reflecting boundary

Algorithms for the simulation of sample paths of Gauss–Markov processes, restricted from below by particular time-dependent reflecting boundaries, are proposed. These algorithms are used to build the histograms of first passage time density through specified boundaries and for the estimation of related moments. Particular attention is dedicated to restricted Wiener and Ornstein–Uhlenbeck processes due to their central role in the class of Gauss–Markov processes

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

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

For the Wiener, Ornstein–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–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

On some integral equations for the evaluation of first-passage-time densities of time-inhomogeneous birth-death processes

New integral equations are proposed to determine first-passage-time densities for time- inhomogeneous birth-death processes. Such equations, particularly suitable for computational purposes, are also used to obtain closed-form expressions for the first-passage-time densities of special birth-death processes of interest in various application fields

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

On a time-inhomogeneous diffusion process with discontinuous drift

We start from a time-inhomogeneous diffusion process, obtained by applying the composition method to two Wiener processes. Then, we investigate the corresponding diffusion process, restricted by a reflecting boundary in the zero-state. Moreover, we consider a special time-inhomogeneous diffusion process symmetric with respect to zero-state characterized by discontinuous drift. Various numerical computations are performed in the presence of periodic noise intensity

On the First-Passage Time Problem for a Feller-Type Diffusion Process

We consider the first-passage time problem for the Feller-type diffusion process, having infinitesimal drift B1(x, t) = α(t) x + β(t) and infinitesimal variance B2(x, t) = 2 r(t)x, defined in the space state [0, +∞), with α(t) ∈ R, β(t) > 0, r(t) > 0 continuous functions. For the time- homogeneous case, some relations between the first-passage time densities of the Feller process and of the Wiener and the Ornstein–Uhlenbeck processes are discussed. The asymptotic behavior of the first-passage time density through a time-dependent boundary is analyzed for an asymptotically constant boundary and for an asymptotically periodic boundary. Furthermore, when β(t) = ξ r(t), with ξ > 0, we discuss the asymptotic behavior of the first-passage density and we obtain some closed-form results for special time-varying boundaries