The Moment Approximation of the First–Passage Time for the Birth–Death Diffusion Process with Immigraton to a Moving Linear Barrier

Basel M. Al-Eideh^1,

¹Department of Quantitative Methods and Information System, Kuwait University, College of Business Administration, Safat, Kuwait

Cite this paper

Basel M. Al-Eideh. The Moment Approximation of the First–Passage Time for the Birth–Death Diffusion Process with Immigraton to a Moving Linear Barrier. American Journal of Applied Mathematics and Statistics. 2015; 3(5):184-189. doi: 10.12691/AJAMS-3-5-2

Abstract

Today, the the development of a mathematical models for population growth of great importance in many fields. The growth and decline of real populations can in many cases be well approximated by the solutions of a stochastic differential equations. However, there are many solutions in which the essentially random nature of population growth should be taken into account. In this paper, we approximating the moments of the first – passage time for the birth and death diffusion process with immigration to a moving linear barriers. This was done by approximating the differential equations by an equivalent difference equations. A simulation study is considered and applied to some values of parameters which showed the capability of the technique.

Keywords

first passage time, birth-death diffusion process, immigration, difference equations

Copyright

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References

[1]	A. J. Alawneh and B. M. Al-Eideh, Moment approximation of the first-passage time for the Ornstein-Uhlenbeck process. Intern. Math. J. Vol.1 (2002), No.3, 255-258.
[2]	B. M. Al-Eideh, First-passage time moment approximation for the Right-Fisher diffusion process with absorbing barriers. Intern. J. Contemp Math. Sciences, Vol.5 (2010), No.27, 1303-1308.
[3]	B. M. Al-Eideh, First-passage time moment approximation for the birth-death diffusion process to a moving linear barriers. J. Stat. & Manag. Systems, Vol.7 (2004), No.1, 173-181.
[4]	I. F. Blake and W. C. Lindsey, Level crossing problems for random processes. IEEE Trans. Information Theory 19 (1973), 295-315.
[5]	D. R. Cox and H. D. Miller, The theory of stochastic processes. Methuen, London (1965).
[6]	D. Darling and A. J. F. Siegert, The first - passage problem for a continuos Markov process. Ann. Math. Statist. 24 (1953), 624-639.
[7]	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.
[8]	W. J. Ewens, Mathematical Population Genetics. Springer-Verlag, Berlin (1979).
[9]	B. Ferebee, The tangent approximation to one-sided Brownian exit densities. Z. Wahrscheinlichkeitsth 61 (1982), 309-326.
[10]	D. L. Iglehart, Limiting diffusion approximation for the many server queueand the repairman problem. J. Appl. Prob. 2 (1965), 429-441.
[11]	S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes. Academic press. New York (1981).
[12]	W.G. Kelly and A.C. Peterson, Difference Equations : An Introduction with Applications. Academic Press, New York (1991).
[13]	D. R. McNeil, Integral functionals of birth and death processes and related limiting distributions. Ann. Math. Statist. 41 (1970), 480-485.
[14]	D. R. McNeil and S. Schach, Central limit analogues for Markov population processes. J. R. Statist. Soc. B, 35 (1973), 1-23.
[15]	M. U. Thomas, Some mean first passage time approximations for the Ornstein – Uhlenbeck process.J. Appl. Prob. 12 (1975), 600-604.
[16]	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.
[17]	A. I. Zeifman, Some estimates of the rate of convergence for birth and death processes. J. Appl. Prob. 28 (1991), 268-277.