×
Beghin, Luisa; Gajda, Janusz; Maheshwari, Aditya

Integro-differential equations linked to compound birth processes with infinitely divisible addends. (English) Zbl 07924811

Math. Methods Appl. Sci. 47, No. 13, 10742-10758 (2024).

MSC:

35R09 Integro-partial differential equations
35R11 Fractional partial differential equations
60G50 Sums of independent random variables; random walks
33E12 Mittag-Leffler functions and generalizations
26A33 Fractional derivatives and integrals

Keywords:

integro-differential equations; convolution-type derivatives; cumulative damage models; first-passage time; infinitely divisible laws
Cite Review PDF
Full Text: DOI arXiv

References:

[1] BeghinL, MacciC. Alternative forms of compound fractional Poisson processes. Abstr Appl Anal. 2012;74750:30. · Zbl 1255.60063
[2] BuchakK, SakhnoL. Compositions of Poisson and gamma processes. Modern Stoch Theory Appl. 2017;4(2):161‐188. · Zbl 1368.60052
[3] SendovaKP, MinkovaLD. Introducing the nonhomogeneous compound‐birth process. Stochastics. 2019:1‐19.
[4] CrowderM. Multivariate survival analysis and competing risks. Taylor and Francis; 2012. · Zbl 1281.62003
[5] DingX, GieseckeK, TomecekP. Time‐changed birth processes and multi‐name credit derivatives. Oper Res. 2009;57:990‐1005. · Zbl 1233.91264
[6] BayraktarE, YangB. Multi‐scale time‐changed birth processes for pricing multi‐name credit derivatives. Appl Math Fin. 2009;16(5):429‐449. · Zbl 1189.91201
[7] SobczykK, SpencerJ. Random Fatigue: From Data to Theory. New York:Academic Press; 1992. · Zbl 0768.73054
[8] OrsingherE, PolitoF. Fractional pure birth process. Bernoulli. 2010;16(3):858‐881. · Zbl 1284.60156
[9] OrsingherE, PolitoF. On a fractional linear birth‐death process. Bernoulli. 2011;17(1):114‐137. · Zbl 1284.60157
[10] CahoyDO, PolitoF. Simulation and estimation for the fractional Yule process. Method Comput Appl Prob. 2012;14(2):383‐403. · Zbl 1253.60090
[11] LanskyP, PolitoF, SacerdoteL. Generalized nonlinear Yule models. J Stat Phys. 2016;165:661‐679. · Zbl 1362.60039
[12] BolandPJ. Statistical and Probabilistic Methods in the Actuarial Science. Boca Raton:Chapman and Hall; 2007. · Zbl 1124.62069
[13] ChernobaiA, BurneckiK, RachevS, TrückS, WeronR. Modelling catastrophe claims with left‐truncated severity distributions. Computational Statistics. 2006;21:537‐555. · Zbl 1164.62411
[14] KostadinovaKY, MinkovaLD. On the Poisson process of order k. Pliska Stud Math Bulgar. 2013;22:117‐128. · Zbl 1374.60091
[15] SchillingRL, SongR, VondracekZ. Bernstein Functions: Theory and Applications, Vol. 37. Berlin:De Gruyter Studies in Mathematics Series; 2010. · Zbl 1197.33002
[16] ToaldoB. Convolution‐type derivatives, hitting‐times of subordinators and time‐changedC_0‐semigroups. Potential Anal. 2015;42(1):115‐140. · Zbl 1315.60050
[17] KochubeiAN. General fractional calculus, evolution equations and renewal processes. Int Eq Oper Theo. 2011;71:583‐600. · Zbl 1250.26006
[18] GajdaJ, WyłomańskaA. Time‐changed Ornstein‐Uhlenbeck process. J Phys A Math Theor. 2015;48:1‐19. · Zbl 1309.82025
[19] OrsingherE, RicciutiC, ToaldoB. On semi‐Markov processes and their Kolmogorov’s integro‐differential equations. Journ Funct Anal. 2018;275(4):830‐868. · Zbl 1396.60094
[20] KilbasAA, SrivastavaHM, TrujilloJJ. Theory and Applications of Fractional Differential Equations, vol. 204 of North‐Holland Mathematics Studies. Amsterdam:Elsevier Science B.V.; 2006. · Zbl 1092.45003
[21] MathaiAM, SaxenaRK, HauboldHJ. The H‐Function: Theory and Applications. New York:Springer; 2010.
[22] GlockleWG, NonnenmacherTF. Fox function representation of non‐Debye relaxation processes. J Stat Phys. 1993;71(3/4):741‐757. · Zbl 0945.82559
[23] MainardiF, MuraA, GorenfloR, StojanovićM. The two forms of fractional relaxation of distributed order. J Vib Cont. 2007;13(9‐10):1249‐1268. · Zbl 1165.26302
[24] BeghinL. Fractional relaxation equations and Brownian crossing probabilities of a random boundary. Adv Appl Probab. 2012;44:479‐505. · Zbl 1251.60032
[25] MillerKS, RossB. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York:John Wiley and Sons; 1993. · Zbl 0789.26002
[26] GorenfloR, KilbasAA, MainardiF, RogosinSV. Mittag‐Leffler Functions, Related Topics and Applications. Berlin Heidelberg:Springer‐Verlag; 2014. · Zbl 1309.33001
[27] CahoyDO, UchaikinVV, WoyczynskiWA. Parameter estimation for fractional Poisson processes. J Statist Plann Inference. 2010;140(11):3106‐3120. · Zbl 1205.62118
[28] DevroyeL. Random variate generation for exponentially and polynomially tilted stable distributions. ACM Trans Model Comput Simul. 2009;19(4):1‐20. · Zbl 1390.65008
[29] HofertM. Sampling exponentially tilted stable distributions. ACM Trans Model Comput Simul. 2011;22(1):3‐11. · Zbl 1386.65051
[30] MaheshwariA, VellaisamyP. Fractional Poisson process time‐changed by Lévy subordinator and its inverse. J Theo Prob. 2019;32(3):1278‐1305. · Zbl 1478.60128
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.