On an integral equation for 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.

A note on first-passage time and some related problems

Expansions for the first-passage-time p.d.f. through a constant boundary and for its Laplace transform are derived in terms of probability currents for a temporally homogeneous diffusion process. Ultimate absorption and recurrence problems are also considered. The moments of the first-passage time are finally explicitly obtained.

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.

A note on first-passage time and some related problems

Expansions for the first-passage-time p.d.f. through a constant boundary and for its Laplace transform are derived in terms of probability currents for a temporally homogeneous diffusion process. Ultimate absorption and recurrence problems are also considered. The moments of the first-passage time are finally explicitly obtained.

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.

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 evaluations and asymptotic approximations of first-passage-time probabilities

The series expansion for the solution of the integral equation for the first-passage-time probability density function, obtained by resorting to the fixed point theorem, is used to achieve approximate evaluations for which error bounds are indicated. A different use of the fixed point theorem is then made to determine lower and upper bounds for asymptotic approximations, and to examine their range of validity.

Calculation of noncrossing probabilities for Poisson processes and its corollaries

The paper describes a new numerical method for the calculation of noncrossing probabilities for arbitrary boundaries by a Poisson process. We find the method to be simple in implementation, quick and efficient - it works reliably for Poisson processes of very high intensity n, up to several thousand. Hence, it can be used to detect unusual features in the finite-sample behaviour of empirical process and trace it down to very high sample sizes. It also can be used as a good approximation for noncrossing probabilities for Brownian motion and Brownian bridge, in particular when the boundaries are not regular. As a numerical example we demonstrate the divergence of normalized Kolmogorov-Smirnov statistics from their prescribed limiting distributions (Eicker (1979), Jaeshke (1979)) for quite large n in contrast to very regular behaviour of statistics of Mason (1983). For the Brownian motion case we considered square-root, Daniels' (1969) and Grooneboom's (1989) boundaries.

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 characterization of the first hitting time of double integral processes to curved boundaries

The problem of finding the probability distribution of the first hitting time of a double integral process (DIP) such as the integrated Wiener process (IWP) has been an important and difficult endeavor in stochastic calculus. It has applications in many fields of physics (first exit time of a particle in a noisy force field) or in biology and neuroscience (spike time distribution of an integrate-and-fire neuron with exponentially decaying synaptic current). The only results available are an approximation of the stationary mean crossing time and the distribution of the first hitting time of the IWP to a constant boundary. We generalize these results and find an analytical formula for the first hitting time of the IWP to a continuous piecewise-cubic boundary. We use this formula to approximate the law of the first hitting time of a general DIP to a smooth curved boundary, and we provide an estimation of the convergence of this method. The accuracy of the approximation is computed in the general case for the IWP and the effective calculation of the crossing probability can be carried out through a Monte Carlo method.

Smoothness of first passage time distributions and a new integral equation for the first passage time density of continuous Markov processes

Let X be a one-dimensional strong Markov process with continuous sample paths. Using Volterra-Stieltjes integral equation techniques we investigate Hölder continuity and differentiability of first passage time distributions of X with respect to continuous lower and upper moving boundaries. Under mild assumptions on the transition function of X we prove the existence of a continuous first passage time density to one-sided differentiable moving boundaries and derive a new integral equation for this density. We apply our results to Brownian motion and its nonrandom Markovian transforms, in particular to the Ornstein-Uhlenbeck process.

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.

Joint Densities of First Hitting Times of a Diffusion Process Through Two Time-Dependent Boundaries

Consider a one-dimensional diffusion process on the diffusion interval I originated in x0 ∈ I. Let a(t) and b(t) be two continuous functions of t, t > t0, with bounded derivatives, a(t) < b(t), and a(t), b(t) ∈ I, for all t > t0. We study the joint distribution of the two random variables Ta and Tb, the first hitting times of the diffusion process through the two boundaries a(t) and b(t), respectively. We express the joint distribution of Ta and Tb in terms of ℙ(Ta < t, Ta < Tb) and ℙ(Tb < t, Ta > Tb), and we determine a system of integral equations verified by these last probabilities. We propose a numerical algorithm to solve this system and we prove its convergence properties. Examples and modeling motivation for this study are also discussed.

Estimates of the Exit Probability for Two Correlated Brownian Motions

Given two correlated Brownian motions (Xt)t≥ 0 and (Yt)t≥ 0 with constant correlation coefficient, we give the upper and lower estimations of the probability ℙ(max0 ≤s≤tXs≥ a, max0 ≤s≤tYs≥ b) for any a,b,t > 0 through explicit formulae. Our strategy is to establish a new reflection principle for two correlated Brownian motions, which can be viewed as an extension of the reflection principle for one-dimensional Brownian motion. Moreover, we also consider the nonexit probability for linear boundaries, i.e. ℙ (Xt ≤ at+c,Yt ≤ bt+d, 0≤ t≤T) for any constants a, b≥0 and c,d, T > 0.