Abstract
In this paper we study the iterated birth process of which we examine the first-passage time distributions and the hitting probabilities. Furthermore, linear birth processes, linear and sublinear death processes at Poisson times are investigated. In particular, we study the hitting times in all cases and examine their long-range behavior. The time-changed population models considered here display upward (birth process) and downward jumps (death processes) of arbitrary size and, for this reason, can be adopted as adequate models in ecology, epidemics and finance situations, under stress conditions.
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References
Alipour, M., Beghin, L., Rostamy, D.: Generalized fractional non-linear birth processes. Method Comput. Appl. Probab. 17(3), 1–16 (2015)
Beghin, L., Orsingher, E.: Poisson process with different Brownian clocks. Stochastics 84(1), 79–112 (2012)
Cahoy, D., Sibatov, R., Uchaikin, V.: Fractional processes: from Poisson to branching one. Int. J. Bifurc. Chaos Appl. Sci. Eng. 18(9), 2717–2725 (2008)
Cahoy, D., Polito, F.: Simulation and estimation for the fractional Yule process. Method Comput. Appl. Probab. 14(2), 383–403 (2012)
Cahoy, D., Polito, F.: Parameter estimation for fractional birth and fractional death processes. Stat. Comput. 24(2), 211–222 (2014)
Di Crescenzo, A., Martinucci, B., Zacks, S.: Compound Poisson process with Poisson subordinator. J. Appl. Probab. 52(2), 360–374 (2015)
Ding, X., Giesecke, K., Tomecek, P.I.: Time-changed birth processes and multi-name credit derivatives. Oper. Res. 57(4), 990–1005 (2009)
Ding, X.: Time-changed birth processes, random thinning, and correlated default risk, Stanford University, PhD thesis (2010)
Donnelly, P., Kurtz, T., Marjoram, P.: Correlation and variability in birth processes. J. Appl. Probab. 30(2), 275–284 (1993)
Gallot, S.F.L.: Absorption and first-passage times for a compound Poisson process in a general upper boundary. J. Appl. Probab. 30(4), 835–850 (1993)
Kumar, A., Nane, E., Vellaisamy, P.: Time-changed Poisson processes. Stat. Probab. Lett. 81(12), 1899–1910 (2011)
Orsingher, E., Polito, F.: Compositions, random sums and continued random fractions of Poisson and fractional Poisson processes. J. Stat. Phys. 148(2), 233–249 (2012)
Orsingher, E., Polito, F., Sakhno, L.: Fractional non-linear, linear and sublinear death processes. J. Stat. Phys. 141(1), 68–93 (2010)
Orsingher, E., Ricciuti, C., Toaldo, B.: Population models at stochastic times, to appear in Adv. Appl. Probab. (2014)
Perry, D., Stadje, W., Zacks, S.: A two-sided first-exit problem for a compound Poisson process with a random upper boundary. Methodol. Comput. Appl. Probab. 7(1), 51–62 (2005)
Riordan, J.: An Introduction to Combinatorial Analysis. Wiley, New York (1958)
Stadje, W., Zacks, S.: Upper first-exit times of compound Poisson processes revisited. Probab. Eng. Inf. Sci. 17(4), 459–465 (2003)
Xu, Y.: First exit times of compound Poisson processes with parallel boundaries. Seq. Anal. Des. Methods Appl. 31(2), 135–144 (2012)
Acknowledgments
We thank Dr. Bruno Toaldo for providing the figures presented in this paper.
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Beghin, L., Orsingher, E. Population Processes Sampled at Random Times. J Stat Phys 163, 1–21 (2016). https://doi.org/10.1007/s10955-016-1475-2
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DOI: https://doi.org/10.1007/s10955-016-1475-2
Keywords
- Yule-Furry process
- Linear and sublinear death processes
- Hitting times
- Extinction probabilities
- First-passage times
- Stirling numbers
- Bell polynomials