First–Passage Time Moment Approximation for the General Diffusion Process to a General Moving Barrier

Basel Mohammad Said Al-Eideh

doi:doi:10.11648/j.ajtas.20180705.11

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First–Passage Time Moment Approximation for the General Diffusion Process to a General Moving Barrier

Basel Mohammad Said Al-Eideh

Published in American Journal of Theoretical and Applied Statistics (Volume 7, Issue 5)

Received: 13 June 2018 Accepted: 9 July 2018 Published: 2 August 2018

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Abstract

The problem of determining the first-passage times to a moving barrier for diffusion and other Markov processes arises in biological modeling, population growth, statistics, engineering, etc. Since the development of mathematical models for population growth of great importance in many fields. Therefore, the growth and decline of real populations can, in many cases, be well approximated by the solutions of stochastic differential equations. However, there are many solutions in which the essentially random nature of population growth should be taken into account. This paper focusses in approximating the moments of the first – passage time for the general diffusion process to a general moving barrier. This was done by approximating the differential equations by equivalent difference equations.

Published in	American Journal of Theoretical and Applied Statistics (Volume 7, Issue 5)
DOI	10.11648/j.ajtas.20180705.11
Page(s)	167-172
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

First Passage Time, General Diffusion Process, Difference Equations, General Moving Barrier

References

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[3]	J. Durbin, Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. J. Appl. Prob. 8 (1971), 431-453.
[4]	D. R. McNeil, Integral functional of birth and death processes and related limiting distributions. Ann. Math. Statist. 41 (1970), 480-485.
[5]	D. L. Iglehart, Limiting diffusion approximation for the many server queue and the repairman problem. J. Appl. Prob. 2 (1965), 429-441.
[6]	D. R. McNeil and S. Schach, Central limit analogues for Markov population processes. J. R. Statist. Soc. B, 35 (1973), 1-23.
[7]	D. R. Cox and H. D. Miller, The theory of stochastic processes. Methuen, London (1965).
[8]	S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes. Academic press. New York (1981).
[9]	M. U. Thomas, Some mean first passage time approximations for the Ornstein – Uhlenbeck process. J. Appl. Prob. 12 (1975), 600-604.
[10]	B. Ferebee, The tangent approximation to one-sided Brownian exit densities. Z. Wahrscheinlichkeitsth 61 (1982), 309-326.
[11]	H. C. Tuchwell and F. Y. M. Wan, First-passage time of Markov processes to moving barriers. J. Appl. Prob. Vol. 21 (1984), 695-709.
[12]	A. J. Alawneh and B. M. Al-Eideh, Moment approximation of the first-passage time for the Ornstien- Uhlenbeck process. Intern. Math. J, Vol. 1 (2002), No. 3, 255-258.
[13]	B. M. Al-Eideh, First-passage time moment approximation for the birth-death diffusion process to a moving linear barrier. J. Stat. & Manag. Systems, Vol. 7 (2004), No. 1, 173-181.
[14]	B. M. Al-Eideh, First-passage time moment approximation for the write-fisher diffusion process with absorbing barrier. Intern. J. Contep. Math, Sciences, Vol. 5 (2010), No. 27, 1303-1308.
[15]	B. M. AL-EIDEH, The Moment Approximation of the First–Passage Time for the Birth–Death Diffusion Process with Immigrations to a Moving Linear Barrier. In Recent Advances in Automation Control, Modeling and Simulation. Edited by Fujita, H., Tuba, M., and Sasaki, J. WSEAS Press, April, (2013), 197-201.
[16]	I. Gikman and A. V. Skorohod, The theory of stochastic processes. Springer-Verlag, Berlin and New York, (1974).
[17]	W. G. Kelly and A. C. Peterson, Difference Equations: An Introduction with Applications. Academic Press, New York (1991).
[18]	A. I. Zeifman, Some estimates of the rate of convergence for birth and death processes. J. Appl. Prob. 28 (1991), 268-277.

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Basel Mohammad Said Al-Eideh. (2018). First–Passage Time Moment Approximation for the General Diffusion Process to a General Moving Barrier. American Journal of Theoretical and Applied Statistics, 7(5), 167-172. https://doi.org/10.11648/j.ajtas.20180705.11

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ACS Style

Basel Mohammad Said Al-Eideh. First–Passage Time Moment Approximation for the General Diffusion Process to a General Moving Barrier. Am. J. Theor. Appl. Stat. 2018, 7(5), 167-172. doi: 10.11648/j.ajtas.20180705.11

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AMA Style

Basel Mohammad Said Al-Eideh. First–Passage Time Moment Approximation for the General Diffusion Process to a General Moving Barrier. Am J Theor Appl Stat. 2018;7(5):167-172. doi: 10.11648/j.ajtas.20180705.11

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@article{10.11648/j.ajtas.20180705.11,
  author = {Basel Mohammad Said Al-Eideh},
  title = {First–Passage Time Moment Approximation for the General Diffusion Process to a General Moving Barrier},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {7},
  number = {5},
  pages = {167-172},
  doi = {10.11648/j.ajtas.20180705.11},
  url = {https://doi.org/10.11648/j.ajtas.20180705.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20180705.11},
  abstract = {The problem of determining the first-passage times to a moving barrier for diffusion and other Markov processes arises in biological modeling, population growth, statistics, engineering, etc. Since the development of mathematical models for population growth of great importance in many fields. Therefore, the growth and decline of real populations can, in many cases, be well approximated by the solutions of stochastic differential equations. However, there are many solutions in which the essentially random nature of population growth should be taken into account. This paper focusses in approximating the moments of the first – passage time for the general diffusion process to a general moving barrier. This was done by approximating the differential equations by equivalent difference equations.},
 year = {2018}
}

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TY  - JOUR
T1  - First–Passage Time Moment Approximation for the General Diffusion Process to a General Moving Barrier
AU  - Basel Mohammad Said Al-Eideh
Y1  - 2018/08/02
PY  - 2018
N1  - https://doi.org/10.11648/j.ajtas.20180705.11
DO  - 10.11648/j.ajtas.20180705.11
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 167
EP  - 172
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20180705.11
AB  - The problem of determining the first-passage times to a moving barrier for diffusion and other Markov processes arises in biological modeling, population growth, statistics, engineering, etc. Since the development of mathematical models for population growth of great importance in many fields. Therefore, the growth and decline of real populations can, in many cases, be well approximated by the solutions of stochastic differential equations. However, there are many solutions in which the essentially random nature of population growth should be taken into account. This paper focusses in approximating the moments of the first – passage time for the general diffusion process to a general moving barrier. This was done by approximating the differential equations by equivalent difference equations.
VL  - 7
IS  - 5
ER  -

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Author Information

Basel Mohammad Said Al-Eideh

Department of Quantitative Methods and Information System, College of Business Administration, Kuwait University, Showaikh, Kuwait

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APA Style

Basel Mohammad Said Al-Eideh. (2018). First–Passage Time Moment Approximation for the General Diffusion Process to a General Moving Barrier. American Journal of Theoretical and Applied Statistics, 7(5), 167-172. https://doi.org/10.11648/j.ajtas.20180705.11

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ACS Style

Basel Mohammad Said Al-Eideh. First–Passage Time Moment Approximation for the General Diffusion Process to a General Moving Barrier. Am. J. Theor. Appl. Stat. 2018, 7(5), 167-172. doi: 10.11648/j.ajtas.20180705.11

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AMA Style

Basel Mohammad Said Al-Eideh. First–Passage Time Moment Approximation for the General Diffusion Process to a General Moving Barrier. Am J Theor Appl Stat. 2018;7(5):167-172. doi: 10.11648/j.ajtas.20180705.11

Copy | Download

@article{10.11648/j.ajtas.20180705.11,
  author = {Basel Mohammad Said Al-Eideh},
  title = {First–Passage Time Moment Approximation for the General Diffusion Process to a General Moving Barrier},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {7},
  number = {5},
  pages = {167-172},
  doi = {10.11648/j.ajtas.20180705.11},
  url = {https://doi.org/10.11648/j.ajtas.20180705.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20180705.11},
  abstract = {The problem of determining the first-passage times to a moving barrier for diffusion and other Markov processes arises in biological modeling, population growth, statistics, engineering, etc. Since the development of mathematical models for population growth of great importance in many fields. Therefore, the growth and decline of real populations can, in many cases, be well approximated by the solutions of stochastic differential equations. However, there are many solutions in which the essentially random nature of population growth should be taken into account. This paper focusses in approximating the moments of the first – passage time for the general diffusion process to a general moving barrier. This was done by approximating the differential equations by equivalent difference equations.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - First–Passage Time Moment Approximation for the General Diffusion Process to a General Moving Barrier
AU  - Basel Mohammad Said Al-Eideh
Y1  - 2018/08/02
PY  - 2018
N1  - https://doi.org/10.11648/j.ajtas.20180705.11
DO  - 10.11648/j.ajtas.20180705.11
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 167
EP  - 172
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20180705.11
AB  - The problem of determining the first-passage times to a moving barrier for diffusion and other Markov processes arises in biological modeling, population growth, statistics, engineering, etc. Since the development of mathematical models for population growth of great importance in many fields. Therefore, the growth and decline of real populations can, in many cases, be well approximated by the solutions of stochastic differential equations. However, there are many solutions in which the essentially random nature of population growth should be taken into account. This paper focusses in approximating the moments of the first – passage time for the general diffusion process to a general moving barrier. This was done by approximating the differential equations by equivalent difference equations.
VL  - 7
IS  - 5
ER  -

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