Representations of the First Hitting Time Density of an Ornstein-Uhlenbeck Process¹

Abstract

Three expressions are provided for the first hitting time density of an Ornstein-Uhlenbeck process to reach a fixed level. The first hinges on an eigenvalue expansion involving zeros of the parabolic cylinder functions. The second is an integral representation involving some special functions whereas the third is given in terms of a functional of a 3-dimensional Bessel bridge. The expressions are used for approximating the density.

A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes

“…In order to improve the estimate in (50), it is an interesting and challenging problem to drop the exponential distribution assumption and consider existing approaches to the p.d.f. of the first-passage time of an OU process for finite values of the boundary s (controlled by M in our problem) [37]. As we can observe in Fig.…”

“…of T s , p Ts (t), known to date are quite involved and for specific applications there is a need to perform numerical computations of the density [36]. As summarized by Alili, Patie and Pedersen in [37], three representations have been proposed for the first-passage time density of an OU-process through a constant boundary. The first expression is a series expansion involving the eigenvalues of a SturmLiouville boundary value problem associated with the Laplace transform of the FPT probability density function.…”

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“…This choice is interesting in that although periods of silence do not occur periodically, the duration between two such silences have a finite mean. Actually, the distribution of excursions shape and duration of the Ornstein-Uhlenbeck process are known in closed form [58]. These distributions are, of course, not heavy tailed: they have exponential tails with exponent α which is the timescale of decay of the process.…”

Bayesian Inference of the Stochastic Gompertz Growth Model for Tumor Growth

“…For example, the estimation methods of parameters (like Newton-Raphson method) require the most-widely used procedures to derive the maximum-likelihood estimate. In addition, there is a floating-point overflow problem in searching the parameter estimates of stochastic diffusion model due to the exponential component (Alili et al, 2005;Linetsky, 2004;Lv and Pitchford, 2007). The advantage of the Bayesian approach using Markov chain Monte Carlo(MCMC) is that the researcher can replace the unobserved variables by simulated variables, relieving the burden of evaluating the likelihood function unconditional to the unobserved variables to allow a focus on the conditional likelihood function.…”