Explicit asymptotics on first passage times of diffusion processes

AbstractWe introduce a unified framework for solving first passage times of time-homogeneous diffusion processes. Using potential theory and perturbation theory, we are able to deduce closed-form truncated probability densities, as asymptotics or approximations to the original first passage time densities, for single-side level crossing problems. The framework is applicable to diffusion processes with continuous drift functions; in particular, for bounded drift functions, we show that the perturbation series converges. In the present paper, we demonstrate examples of applying our framework to the Ornstein–Uhlenbeck, Bessel, exponential-Shiryaev, and hypergeometric diffusion processes (the latter two being previously studied by Dassios and Li (2018) and Borodin (2009), respectively). The purpose of this paper is to provide a fast and accurate approach to estimating first passage time densities of various diffusion processes.

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Spectral Theory for Skip-Free Markov Chains

Probability in the Engineering and Informational Sciences ◽

10.1017/s026996480000098x ◽

1989 ◽

Vol 3 (1) ◽

pp. 77-88 ◽

Cited By ~ 9

Author(s):

Joseph Abate ◽

Ward Whitt

Keyword(s):

Markov Chains ◽

Spectral Theory ◽

First Passage Time ◽

Passage Time ◽

First Passage Times ◽

Simple Relationship ◽

First Passage ◽

Birth And Death Processes ◽

The Laplace Transform ◽

Passage Times

The distribution of upward first passage times in skip-free Markov chains can be expressed solely in terms of the eigenvalues in the spectral representation, without performing a separate calculation to determine the eigenvectors. We provide insight into this result and skip-free Markov chains more generally by showing that part of the spectral theory developed for birth-and-death processes extends to skip-free chains. We show that the eigenvalues and eigenvectors of skip-free chains can be characterized in terms of recursively defined polynomials. Moreover, the Laplace transform of the upward first passage time from 0 to n is the reciprocal of the nth polynomial. This simple relationship holds because the Laplace transforms of the first passage times satisfy the same recursion as the polynomials except for a normalization.

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On first passage times of sticky reflecting diffusion processes with double exponential jumps

Journal of Applied Probability ◽

10.1017/jpr.2019.93 ◽

2020 ◽

Vol 57 (1) ◽

pp. 221-236 ◽

Cited By ~ 1

Author(s):

Shiyu Song ◽

Yongjin Wang

Keyword(s):

Diffusion Processes ◽

First Passage Time ◽

Passage Time ◽

Explicit Solutions ◽

First Passage ◽

Uhlenbeck Process ◽

Double Exponential ◽

Upper Level ◽

First Passage Problem ◽

Passage Times

AbstractWe explore the first passage problem for sticky reflecting diffusion processes with double exponential jumps. The joint Laplace transform of the first passage time to an upper level and the corresponding overshoot is studied. In particular, explicit solutions are presented when the diffusion part is driven by a drifted Brownian motion and by an Ornstein–Uhlenbeck process.

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First-passage times with PFr densities

Journal of Applied Probability ◽

10.2307/3213758 ◽

1985 ◽

Vol 22 (1) ◽

pp. 185-196 ◽

Cited By ~ 27

Author(s):

David Assaf ◽

Moshe Shared ◽

J. George shanthikumar

Keyword(s):

First Passage Time ◽

Compound Poisson Process ◽

Passage Time ◽

First Passage Times ◽

Embedded Markov Chain ◽

First Passage ◽

Completely Monotone ◽

Totally Positive ◽

Finite State ◽

Passage Times

It is shown that if a finite-state continuous-time Markov process can be uniformized such that the embedded Markov chain has a TPr (totally positive of order r) transition matrix, then the first-passage time from state 0 to any other state has a PFr (Polya frequency of order r) density. As a consequence, results of Keilson (1971), Esary, Marshall and Proschan (1973), Ghosh and Ebrahimi (1982) and Derman, Ross and Schechner (1983) are strengthened. It is also shown that some cumulative damage shock models, with an underlying compound Poisson process and ‘damages' which are not necessarily non-negative, are associated with wear processes having PFr first-passage times to any threshold. First-passage times with completely monotone densities are also discussed.

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A symmetry-based constructive approach to probability densities for one-dimensional diffusion processes

Journal of Applied Probability ◽

10.2307/3214376 ◽

1989 ◽

Vol 26 (4) ◽

pp. 707-721 ◽

Cited By ~ 12

Author(s):

V. Giorno ◽

A. G. Nobile ◽

L. M. Ricciardi

Keyword(s):

Diffusion Processes ◽

First Passage Time ◽

Passage Time ◽

Constructive Approach ◽

First Passage ◽

One Dimensional ◽

Dimensional Diffusion ◽

Probability Densities ◽

Reflecting Boundaries ◽

Natural Boundaries

Special symmetry conditions on the transition p.d.f. of one-dimensional time-homogeneous diffusion processes with natural boundaries are investigated and exploited to derive closed-form results concerning the transition p.d.f.'s in the presence of absorbing and reflecting boundaries and the first-passage-time p.d.f. through time-dependent boundaries.

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The Protein Hourglass: Exact First Passage Time Distributions for Protein Thresholds

10.1101/2020.08.14.251223 ◽

2020 ◽

Author(s):

Krishna Rijal ◽

Ashok Prasad ◽

Dibyendu Das

Keyword(s):

First Passage Time ◽

Self Regulation ◽

Threshold Value ◽

Passage Time ◽

First Passage Times ◽

Lambda Phage ◽

First Passage ◽

Analytical Expression ◽

Protein Levels ◽

Passage Times

Protein thresholds have been shown to act as an ancient timekeeping device, such as in the time to lysis of E. coli infected with bacteriophage lambda. The time taken for protein levels to reach a particular threshold for the first time is defined as the first passage time of the protein synthesis system, which is a stochastic quantity. The first few moments of the distribution of first passage times were known earlier, but an analytical expression for the full distribution was not available. In this work, we derive an analytical expression for the first passage times for a long-lived protein. This expression allows us to calculate the full distribution not only for cases of no self-regulation, but also for both positive and negative self-regulation of the threshold protein. We show that the shape of the distribution matches previous experimental data on lambda-phage lysis time distributions. We also provide analytical expressions for the FPT distribution with non-zero degradation in Laplace space. Furthermore, we study the noise in the precision of the first passage times described by coefficient of variation (CV) of the distribution as a function of the protein threshold value. We show that under conditions of positive self-regulation, the CV declines monotonically with increasing protein threshold, while under conditions of linear negative self-regulation, there is an optimal protein threshold that minimizes the noise in the first passage times.

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Ageing first-passage times of Markov processes: a matrix approach

Journal of Applied Probability ◽

10.2307/3215169 ◽

1997 ◽

Vol 34 (1) ◽

pp. 1-13 ◽

Cited By ~ 11

Author(s):

Haijun Li ◽

Moshe Shaked

Keyword(s):

Markov Process ◽

Markov Processes ◽

First Passage Time ◽

Passage Time ◽

Critical Value ◽

First Passage Times ◽

Matrix Approach ◽

First Passage ◽

Passage Times

Using a matrix approach we discuss the first-passage time of a Markov process to exceed a given threshold or for the maximal increment of this process to pass a certain critical value. Conditions under which this first-passage time possesses various ageing properties are studied. Some results previously obtained by Li and Shaked (1995) are extended.

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

Journal of Applied Probability ◽

10.1017/s0021900200097084 ◽

1983 ◽

Vol 20 (01) ◽

pp. 197-201 ◽

Cited By ~ 9

Author(s):

L. M. Ricciardi ◽

S. Sato

Keyword(s):

Wiener Process ◽

Diffusion Processes ◽

First Passage Time ◽

Passage Time ◽

First Passage ◽

Probability Densities

A procedure is indicated to estimate first-passage-time p.d.f.'s through varying boundaries for a class of diffusion processes that can be transformed into the Wiener process by rather general transformations. Although this procedure is adapted to Durbin's [4] algorithm, it could be extended to other existing computation methods.

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Mean first-passage times of Brownian motion and related problems

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1952.0051 ◽

1952 ◽

Vol 211 (1106) ◽

pp. 431-443 ◽

Cited By ~ 42

Keyword(s):

Brownian Motion ◽

First Passage Time ◽

Passage Time ◽

First Passage Times ◽

Analytical Relationship ◽

First Passage ◽

Special Boundaries ◽

Mean First Passage Times ◽

Special Cases ◽

Passage Times

The theory of first-passage times of Brownian motion is developed in general, and it is shown that for certain special boundaries—the only ones of any importance—mean first-passage times can be derived very simply, avoiding the usual method involving series. Moreover, these formulae have a close analytical relationship to the better-known type of formulae for average 'displacements’ in given intervals; there exist certain pairs of reciprocal relations. Some new formulae, of mathematical interest, for translational Brownian motion are given. The main application of the general theory, however, lies in the derivation of experimentally particularly useful formulae for rotational Brownian motion. Special cases when external forces are present, and mean reciprocal first-passage times are discussed briefly, and finally it is shown how finite times of observation modify the mean first-passage time formulae of free Brownian motion.

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On the evaluation of first-passage-time probability densities via non-singular integral equations

Advances in Applied Probability ◽

10.2307/1427196 ◽

1989 ◽

Vol 21 (1) ◽

pp. 20-36 ◽

Cited By ~ 51

Author(s):

V. Giorno ◽

A. G. Nobile ◽

L. M. Ricciardi ◽

S. Sato

Keyword(s):

Diffusion Processes ◽

First Passage Time ◽

Singular Integral Equations ◽

Passage Time ◽

Time Dependent ◽

Computational Results ◽

First Passage ◽

One Dimensional ◽

Dimensional Diffusion ◽

Probability Densities

The algorithm given by Buonocore et al. [1] to evaluate first-passage-time p.d.f.’s for Wiener and Ornstein–Uhlenbeck processes through a time-dependent boundary is extended to a wide class of time-homogeneous one-dimensional diffusion processes. Several examples are thoroughly discussed along with some computational results.

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