×

zbMATH — the first resource for mathematics

Large claims approximations for risk processes in a Markovian environment. (English) Zbl 0814.60067
The paper provides some results concerning a risk process with initial reserve \(u\), which evolves in a finite Markovian environment with initial state \(i\). Denoting by \(\beta_ j\) the arrival intensity and by \(B_ j\) the claim size distribution when the environment is in the state \(j( \in E)\), the authors suppose that there is a subset \(E^{(1)}\) of \(E\) such that, for every \(j \in E^{(1)}\), \(B_ j\) verify the asymptotic condition \(1-B_ j(x) \sim b_ j (1 - H(x))\), as \(x \to \infty\), for some \(b_ j \in (0, + \infty)\) and some p.d.f. \(H\), whose tail is a subexponential density, and, for every \(j \in E \backslash E^{(1)}\), \(1 - B_ j(x) = o(1 - H(x))\). Under these hypotheses, denoting by \(\psi_ i (u)\) the probability of ruin for the above mentioned risk process, the authors prove that \(\psi_ i (u) \sim c_ i \int^ \infty_ u (1 - H(x))dx\), for some explicitly computed constant \(c_ i\). In the last part of the paper it is proved that similar results hold for the tail of the waiting time in a Markov-modulated \(M/G/1\) queue whenever the service times satisfy similar hypotheses.

MSC:
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
60K25 Queueing theory (aspects of probability theory)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abate, J.; Choudhury, G.L.; Whitt, W., Waiting time probabilities in queues with long-tail service time distribution, Queueing systems, (1994), to appear in: · Zbl 0805.60097
[2] Asmussen, S., Risk theory in a Markovian environment, Scand. actuar. J., 69-100, (1989) · Zbl 0684.62073
[3] Asmussen, S., Ladder heights and the Markov-modulated M/G/1 queue, Stoch. proc. appl., 37, 313-326, (1991) · Zbl 0734.60091
[4] S. Asmussen, Ruin Probabilities, Book manuscript, Aalborg University
[5] Asmussen, S.; Rolski, T., Computational methods in risk theory: a matrix-algorithmic approach, Insurance math. econom., 10, 259-274, (1991) · Zbl 0748.62058
[6] Asmussen, S.; Rolski, T., Risk theory in a periodic environment: the cramér-lundberg approximation and lundbergs inequality, Math. oper. res., 19, 410-433, (1994) · Zbl 0801.60091
[7] Athreya, K.B.; Ney, P., Branching processes, (1972), Springer New York · Zbl 0259.60002
[8] Bingham, N.H.; Goldie, C.M.; Teugels, J.L., Regular variation, (1987), Cambridge University Press Cambridge · Zbl 0617.26001
[9] Björk, T.; Grandell, J., Exponential inequalities for ruin probabilities in the Cox case, Scand. actuar. J., 77-111, (1988) · Zbl 0668.62072
[10] Cline, D.B.H., Convolution tails, product tails and domains of attraction, Probab. theory related fields, 72, 529-557, (1986) · Zbl 0577.60019
[11] Cohen, J.W., The single server queue, (1982), North-Holland Amsterdam · Zbl 0481.60003
[12] Embrechts, P.; Goldie, C.M.; Veraverbeke, N., Subexponentiality and infinite divisibility, Z. wahr. verw. gebiete, 49, 335-347, (1979) · Zbl 0397.60024
[13] Embrechts, P.; Grandell, J.; Schmidli, H., Finite-time lundberg inequalities in the Cox case, Scand. actuar. J., (1993) · Zbl 0785.62094
[14] Embrechts, P.; veraverbeke, N., Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance math. econom., 1, 55-72, (1982) · Zbl 0518.62083
[15] Embrechts, P.; Villaseñor, J.A., Ruin estimates for large claims, Insurance math. econom., 7, 269-274, (1988) · Zbl 0666.62098
[16] Feller, W., An introduction to probability theory and its applications, Vol. 2, (1971), Wiley New York · Zbl 0219.60003
[17] Henriksen, L.Fløe, Large claims in ruin probability theory — simulation studies and the case of a Markovian environment, (), (in Danish)
[18] Grandell, J., Aspects of risk theory, (1990), Springer Berlin
[19] Janssen, J., Some transient results on the M/SM/1 special semi-Markov model in risk and queueing theories, Astin bull., 11, 41-51, (1980)
[20] Janssen, J.; Reinhard, J.M., Probabilités de ruine pour une classe de modeles de risque semi-markoviens, Astin bull., 15, 123-133, (1985)
[21] Klüppelberg, C., Subexponential distributions and integrated tails, J. appl. probab., 25, 132-141, (1988) · Zbl 0651.60020
[22] Klüppelberg, C., Estimation of ruin probabilities by means of hazard rates, Insurance math. econom., 8, 279-285, (1989) · Zbl 0686.62093
[23] Moler, C.; Loan, C.Van, Nineteen dubious ways to compute the exponential of a matrix, SIAM rev., 20, 801-836, (1978) · Zbl 0395.65012
[24] Reinhard, J.M., On a class of semi-Markov risk models obtained as classical risk models in a Markovian environment, Astin bull., XIV, 23-43, (1984)
[25] Sengupta, B., Markov processes whose steady-state distribution is matrix-exponential with an application to the GI/PH/1 queue, Adv. appl. probab., 21, 159-180, (1989) · Zbl 0672.60090
[26] Bahr, B.von, Asymptotic ruin probabilities when exponential moments do not exist, Scand. actuar. J., 6-10, (1975) · Zbl 0321.62102
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.