CONTINUOUS TIME INTERPOLATION OF MONOTONE MARKED RANDOM MEASURES WITH APPLICATIONS

TitleCONTINUOUS TIME INTERPOLATION OF MONOTONE MARKED RANDOM MEASURES WITH APPLICATIONS
Publication TypeJournal Article
Year of Publication2018
AuthorsDSHALALOW JEWGENI, NANDYOSE KIZZA
JournalNeural, Parallel, and Scientific Computations
Volume26
Issue2
Start Page119
Pagination24
Date Published2018
ISSN1056-2176
Keywords60G25, 60G50, 60G51, 60G52, 60G55, 60G57, 60K05, 60K35, 60K40, 90B10, 90B15, 90B18, 90B25
Abstract

We study a class of monotone delayed marked point processes that model stochastic networks (under attacks), status of queueing systems during vacation modes, responses to cancer treatments (such as chemotherapy and radiation), hostile ambushes in economics and warfare. We are interested in the behavior of such a process about a fixed threshold. It presents an analytic challenge, because of the arbitrary nature of random marks. We target the first passage time, pre-first passage time, the status of the associated continuous time parameter process between these two epochs, and the status of the process upon these two epochs. A joint functional of these stochastic quantities is investigated in the transient mode. Analytically tractable formulas are obtained and demonstrated on special cases of marked Poisson processes.

URLhttps://acadsol.eu/npsc/articles/26/2/1.pdf
DOI10.12732/npsc.v26i2.1
Refereed DesignationRefereed
Full Text

REFERENCES
[1] Agarwal, R.P., Dshalalow, J.H., and O’Regan, D., Time sensitive functionals of
marked Cox processes, Journ. of Math. Analysis and Appl., 293 (2004), 14-27.
[2] Al-Matar, N. and Dshalalow, J.H., A game-theoretic approach in single-server
queues with maintenance. Time sensitive analysis, Commun. in Appl. Nonlin. Analysis, 17, No. 1 (2010), 65-92.
[3] Al-Matar, N. and Dshalalow, J.H., Time sensitive functionals in classes of queues
with sequential maintenance, Stochastic Models, 27 (2011), 687–704.
[4] Atiya, A.F. and Metwally, S.A., Efficient estimation of first passage time density
function for jump-diffusion processes, SIAM J. Sci. Comput., 26(5), 1760–1775.
[5] Aurzada, F., Iksanov, A., and Meiners, M., Exponential moments of first passage
times and related quantities for L´evy processes, Mathematische Nachrichten
(2015), (DOI: 10.1002/mana.201400289).
[6] Bingham, N.H., Random walk and fluctuation theory, in Handbook of Statistics
(Eds. D.N. Shanbhag and C.R. Rao), Volume 19, 2001, Elsevier Science, 171-213.
[7] Borokov, A.A., On the first passage time for one class of processes with independent
increments. Theor. Probab. Appl., 10 (1964), 331–334.
[8] Coutin, L. and Dorobantu, D., First passage time law for some Levy processes
with compound Poisson: Existence of a density, Bernoulli, 17:4 (2011), 1127- 1135.
[9] Crescenzo, A.D. and Martinucci, B., On a first passage-time problem for the
compound power-law process, Stochastic Models, 25 (2009), 420-435.
[10] Dshalalow, J.H., First excess level of v
[19] Dshalalow, J.H., Iwezulu, K., and White, R.T., Discrete operational calculus
in delayed stochastic games, Neural, Parallel, and Scientific Computations, 24 (2016), 55-64.
[20] Dshalalow, J.H. and Ke, H.-J., Multilayers in a modulated stochastic game,
Journ. of Math. Analysis and Applications, 353 (2009), 553-565.
[21] Dshalalow, J.H. and Liew, A., On exit time of a multivariate random walk and
its embedding in a quasi-Poisson process, Stochastic Analysis and Applications,
24 (2006), 451-474.
[22] Dshalalow, J.H. and White, R.T., On reliability of stochastic networks, Neural,
Parallel, and Scientific Computations, 21 (2013) 141-160.
[23] Dshalalow, J.H. and White, R.T., On strategic defense in stochastic networks,
Stochastic Analysis and Applications, 32 (2014), 365–396.
[24] Dshalalow, J.H. and White, R.T., Time sensitive analysis of independent and stationary
increment processes, Journal of Mathematical Analysis and Applications
443 (2016), 817-833.
[25] Hida, T. (Editor), Mathematical Approach to Fluctuations: Astronomy, Biology
and Quantum Dynamics: Proceedings of the Iias Workshop: Kyoto, Japan, May
18-21, 1992, World Scientific Publishers, 1995).
[26] Fishburn, P.C., Non-cooperative stochastic dominance games, International Journal
of Game Theory, 7:1 (1978), 51-61.
[27] Kadankova, T.V., Exit, passage, and crossing times and overshoots for a Poisson
compound process with an exponential component, Theor. Probability and
Math. Statist., 75 (2007), 23-29.
[28] Kyprianou, A.E. and Pistorius, M.R., Perpetual options and Canadization through
fluctuation theory, Ann. Appl. Prob., 13:3 (2003), 1077-1098.
[29] Mellander, E., Vredin, A, and Warne, A., Stochastic trends and economic fluctuations
in a small open economy, J. Applied Econom., 7:4 (1992), 369-94.
[30] Muzy, J., Delour1, J., and Bacry, E., Modelling fluctuations of financial time
series: from cascade process to stochastic volatility model, Eur. Phys. J. B 17 (2000), 537-548.
[31] Novikov, A, Melchers, R.E., Shinjikashvili, E., and Kordzakhia, N., First passage
time of filtered Poisson process with exponential shape function, Probabilistic
Engineering Mechanics, 20:1 (2005), 57-65
[32] Redner, S., A Guide to First-Passage Processes, Cambridge University Press,
Cambridge, 2001.
[33] Shinozuka, M. and Wu, W-F., On the first passage problem and its application to
earthquake engineering, Proceedings of Ninth World Conference on Earthquake
Engineering, August 2-9, (VIII) 1988, Tokyo-Kyoto, Japan.
[34] Tak´acs, L., On fluctuations of sums of random variables, in Studies in Probability
and Ergodic Theory. Advances in Mathematics; Supplementary Studies, Volume
2, ed. by G.-C. Rota, (1978), 45-93.
[35] Yin, C., Wen, Y., Zong, Z., and Shen, Y., The first passage time problem for
mixed-exponential jump processes with applications in insurance and finance,
Abstract and Applied Analysis (2014), 9 pages.
[36] Zolotarev, V.M. The first passage time of a level and the behavior at infnity for a
class of processes with independent increments, Theor. Probab. Appl, 9 (1964)
653–664.