A note on the evaluation of first-passage-time probability densities

Exponential trends of Ornstein–Uhlenbeck first-passage-time densities

The asymptotic behaviour of the first-passage-time p.d.f. through a constant boundary for an Ornstein–Uhlenbeck process is investigated for large boundaries. It is shown that an exponential p.d.f. arises, whose mean is the average first-passage time from 0 to the boundary. The proof relies on a new recursive expression of the moments of the first-passage-time p.d.f. The excellent agreement of theoretical and computational results is pointed out. It is also remarked that in many cases the exponential behaviour actually occurs even for small values of time and boundary.

Exponential trends of Ornstein–Uhlenbeck first-passage-time densities

The asymptotic behaviour of the first-passage-time p.d.f. through a constant boundary for an Ornstein–Uhlenbeck process is investigated for large boundaries. It is shown that an exponential p.d.f. arises, whose mean is the average first-passage time from 0 to the boundary. The proof relies on a new recursive expression of the moments of the first-passage-time p.d.f. The excellent agreement of theoretical and computational results is pointed out. It is also remarked that in many cases the exponential behaviour actually occurs even for small values of time and boundary.

On the probability densities of an Ornstein–Uhlenbeck process with a reflecting boundary

We show that the transition p.d.f. of the Ornstein–Uhlenbeck process with a reflection condition at an assigned state S is related by integral-type equations to the free transition p.d.f., to the transition p.d.f. in the presence of an absorption condition at S, to the first-passage-time p.d.f. to S and to the probability current. Such equations, which are also useful for computational purposes, yield as an immediate consequence all known closed-form results for Wiener and Ornstein–Uhlenbeck processes.

On an integral equation for first-passage-time probability densities

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ∼ 2, a, b E ∼) for which no explicit analytical results have previously been available.

A note on the Volterra integral equation for the first-passage-time probability density

In this paper we prove the validity of the Volterra integral equation for the evaluation of first-passage-time probability densities through varying boundaries, given by Buonocore et al. [1], for the case of diffusion processes not necessarily time-homogeneous. We study, specifically those processes that can be obtained from the Wiener process in the sense of [5]. A study of the kernel of the integral equation, in the same way as that by Buonocore et al. [1], is done. We obtain the boundaries for which closed-form solutions of the integral equation, without having to solve the equation, can be obtained. Finally, a few examples are given to indicate the actual use of our method.

On some time-non-homogeneous diffusion approximations to queueing systems

Time-non-homogeneous diffusion approximations to single server–single queue–FCFS discipline systems are considered. Under various assumptions on the nature of the time-dependent functions appearing in the infinitesimal moments the transient and the regime behaviour of the approximating diffusions are analysed in some detail. Special attention is then given to the study of a diffusion approximation characterized by a linear drift and by a periodically time-varying infinitesimal variance. Unlike the behaviour of transition functions and moments, the p.d.f. of the busy period is seen to be unaffected by the presence of such periodicity.

On some time-non-homogeneous diffusion approximations to queueing systems

Time-non-homogeneous diffusion approximations to single server–single queue–FCFS discipline systems are considered. Under various assumptions on the nature of the time-dependent functions appearing in the infinitesimal moments the transient and the regime behaviour of the approximating diffusions are analysed in some detail. Special attention is then given to the study of a diffusion approximation characterized by a linear drift and by a periodically time-varying infinitesimal variance. Unlike the behaviour of transition functions and moments, the p.d.f. of the busy period is seen to be unaffected by the presence of such periodicity.

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.

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.