Weak convergence of first passage time processes

Ward Whitt

doi:10.2307/3211913

Weak convergence of first passage time processes

Journal of Applied Probability ◽

10.1017/s0021900200035440 ◽

1971 ◽

Vol 8 (02) ◽

pp. 417-422 ◽

Cited By ~ 4

Author(s):

Ward Whitt

Keyword(s):

Stochastic Process ◽

Weak Convergence ◽

First Passage Time ◽

Continuous Mapping ◽

Passage Time ◽

Continuous Functions ◽

First Passage ◽

Image Position ◽

Supremum Process

LetD = D[0, ∞) be the space of all real-valued right-continuous functions on [0, ∞) with limits from the left. For any stochastic processXinD,let the associatedsupremum processbeS(X), wherefor anyx ∊ D. It is easy to verify thatS:D→Dis continuous in any of Skorohod's (1956) topologies extended fromD[0,1] toD[0, ∞) (cf. Stone (1963) and Whitt (1970a, c)). Hence, weak convergenceXn⇒XinDimplies weak convergenceS(Xn) ⇒S(X) inDby virtue of the continuous mapping theorem (cf. Theorem 5.1 of Billingsley (1968)).

Download Full-text

First-passage-time densities for time-non-homogeneous diffusion processes

Journal of Applied Probability ◽

10.2307/3215089 ◽

1997 ◽

Vol 34 (3) ◽

pp. 623-631 ◽

Cited By ~ 30

Author(s):

R. Gutiérrez ◽

L. M. Ricciardi ◽

P. Román ◽

F. Torres

Keyword(s):

Diffusion Processes ◽

First Passage Time ◽

Passage Time ◽

Continuous Functions ◽

Probability Density Functions ◽

Density Functions ◽

First Passage ◽

One Dimensional ◽

Dimensional Diffusion ◽

Natural Boundaries

In this paper we study a Volterra integral equation of the second kind, including two arbitrary continuous functions, in order to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous one-dimensional diffusion processes with natural boundaries. These results generalize those which were obtained for time-homogeneous diffusion processes by Giorno et al. [3], and for some particular classes of time-non-homogeneous diffusion processes by Gutiérrez et al. [4], [5].

Download Full-text

Fractional compound Poisson processes with multiple internal states

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2018001 ◽

2018 ◽

Vol 13 (1) ◽

pp. 10 ◽

Cited By ~ 8

Author(s):

Pengbo Xu ◽

Weihua Deng

Keyword(s):

Stochastic Process ◽

Random Walk ◽

Waiting Time ◽

Anomalous Diffusion ◽

First Passage Time ◽

Passage Time ◽

Poisson Processes ◽

First Passage ◽

Internal States ◽

Functional Distribution

For the particles undergoing the anomalous diffusion with different waiting time distributions for different internal states, we derive the Fokker-Planck and Feymann-Kac equations, respectively, describing positions of the particles and functional distributions of the trajectories of particles; in particular, the equations governing the functional distribution of internal states are also obtained. The dynamics of the stochastic processes are analyzed and the applications, calculating the distribution of the first passage time and the distribution of the fraction of the occupation time, of the equations are given. For the further application of the newly built models, we make very detailed discussions on the none-immediately-repeated stochastic process, e.g., the random walk of smart animals.

Download Full-text

On the First-Passage Distribution for the Envelope of a Nonstationary Narrow-Band Stochastic Process

Journal of Applied Mechanics ◽

10.1115/1.3423390 ◽

1974 ◽

Vol 41 (3) ◽

pp. 793-797 ◽

Cited By ~ 31

Author(s):

W. C. Lennox ◽

D. A. Fraser

Keyword(s):

Stochastic Process ◽

Hypergeometric Functions ◽

Narrow Band ◽

First Passage Time ◽

Wide Band ◽

Passage Time ◽

First Passage ◽

Eigenvalues And Eigenfunctions ◽

One Dimensional ◽

Backward Equation

A narrow-band stochastic process is obtained by exciting a lightly damped linear oscillator by wide-band stationary noise. The equation describing the envelope of the process is replaced, in an asymptotic sense, by a one-dimensional Markov process and the modified Kolmogorov (backward) equation describing the first-passage distribution function is solved exactly using classical methods by reducing the problem to that of finding the related eigenvalues and eigenfunctions; in this case degenerate hypergeometric functions. If the exciting process is white noise, the analysis is exact. The method also yields reasonable approximations for the first-passage time of the actual narrow-band process for either a one-sided or a symmetric two-sided barrier.

Download Full-text

First Passage Time Analysis of Stochastic Process Algebra Using Partial Orders

Tools and Algorithms for the Construction and Analysis of Systems - Lecture Notes in Computer Science ◽

10.1007/3-540-45319-9_16 ◽

2001 ◽

pp. 220-235 ◽

Cited By ~ 5

Author(s):

Theo C. Ruys ◽

Rom Langerak ◽

Joost-Pieter Katoen ◽

Diego Latella ◽

Mieke Massink

Keyword(s):

Stochastic Process ◽

Process Algebra ◽

First Passage Time ◽

Passage Time ◽

Partial Orders ◽

Time Analysis ◽

First Passage ◽

Stochastic Process Algebra

Download Full-text

A limit theorem for random walks with drift

Journal of Applied Probability ◽

10.2307/3212307 ◽

1967 ◽

Vol 4 (1) ◽

pp. 144-150 ◽

Cited By ~ 11

Author(s):

C. C. Heyde

Keyword(s):

Random Walk ◽

Limit Theorem ◽

Random Walks ◽

First Passage Time ◽

Random Variables ◽

Passage Time ◽

Random Walk Process ◽

First Passage ◽

Time Out ◽

Image Position

Let Xi, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables. Write and for x ≧ 0 define M(x) + 1 is then the first passage time out of the interval (– ∞, x] for the random walk process Sn.

Download Full-text

A new integral equation for the evaluation of first-passage-time probability densities

Advances in Applied Probability ◽

10.1017/s0001867800017432 ◽

1987 ◽

Vol 19 (04) ◽

pp. 784-800 ◽

Cited By ~ 23

Author(s):

A. Buonocore ◽

A. G. Nobile ◽

L. M. Ricciardi

Keyword(s):

Integral Equation ◽

Diffusion Processes ◽

First Passage Time ◽

Numerical Procedure ◽

Passage Time ◽

Continuous Functions ◽

Time Dependent ◽

First Passage ◽

One Dimensional ◽

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.

Download Full-text

A limit theorem for random walks with drift

Journal of Applied Probability ◽

10.1017/s0021900200025304 ◽

1967 ◽

Vol 4 (01) ◽

pp. 144-150 ◽

Cited By ~ 3

Author(s):

C. C. Heyde

Keyword(s):

Random Walk ◽

Limit Theorem ◽

Random Walks ◽

First Passage Time ◽

Random Variables ◽

Passage Time ◽

Random Walk Process ◽

First Passage ◽

Time Out ◽

Image Position

Let Xi , i = 1, 2, 3, … be a sequence of independent and identically distributed random variables. Write and for x ≧ 0 define M(x) + 1 is then the first passage time out of the interval (– ∞, x] for the random walk process Sn.

Download Full-text

A new integral equation for the evaluation of first-passage-time probability densities

Advances in Applied Probability ◽

10.2307/1427102 ◽

1987 ◽

Vol 19 (4) ◽

pp. 784-800 ◽

Cited By ~ 124

Author(s):

A. Buonocore ◽

A. G. Nobile ◽

L. M. Ricciardi

Keyword(s):

Integral Equation ◽

Diffusion Processes ◽

First Passage Time ◽

Numerical Procedure ◽

Passage Time ◽

Continuous Functions ◽

Time Dependent ◽

First Passage ◽

One Dimensional ◽

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.

Download Full-text

A Damped Telegraph Random Process with Logistic Stationary Distribution

Journal of Applied Probability ◽

10.1017/s0021900200006410 ◽

2010 ◽

Vol 47 (01) ◽

pp. 84-96 ◽

Cited By ~ 10

Author(s):

Antonio Di Crescenzo ◽

Barbara Martinucci

Keyword(s):

Stochastic Process ◽

Real Line ◽

First Passage Time ◽

Passage Time ◽

First Passage ◽

Finite Velocity ◽

Telegraph Process ◽

Velocity Changes ◽

Random Times ◽

Special Case

We introduce a stochastic process that describes a finite-velocity damped motion on the real line. Differently from the telegraph process, the random times between consecutive velocity changes have exponential distribution with linearly increasing parameters. We obtain the probability law of the motion, which admits a logistic stationary limit in a special case. Various results on the distributions of the maximum of the process and of the first passage time through a constant boundary are also given.

Download Full-text