zbMATH — the first resource for mathematics

Some results about the expected ruin time in Markov-modulated risk models. (English) Zbl 0859.60081
Summary: We investigated the expected ruin time of Markov-modulated risk models. It turns out that the expected ruin time \(\xi(u)\), depending on the initial risk reserve \(u\in\mathbb{R}^+\), is asymptotically linear. In the two-state model we are able to derive exact formulas. A very interesting result is the monotonicity property of \(\xi(u)\). We show that the more slowly the environment changes, the greater is the expected ruin time.

60K10 Applications of renewal theory (reliability, demand theory, etc.)
91B30 Risk theory, insurance (MSC2010)
Full Text: DOI
[1] Asmussen, S., Risk theory in a Markovian environment, Scandinavian actuarial journal, 69-100, (1989) · Zbl 0684.62073
[2] Asmussen, S.; Frey, A.; Rolski, T.; Schmidt, V., Does Markov-modulation increase the risk?, ASTIN bulletin, 25, no. 1, 49-66, (1995)
[3] Bäuerle, N., Stochastic models with a Markovian environment, (), (in German)
[4] Bäuerle, N., Monotonicity results for MR/GI/1 queues, Journal of applied probability, 35, (1997), To appear in · Zbl 0885.60081
[5] Chang, C.; Chao, X.; Pinedo, M., Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross’s conjecture, Advances of applied probability, 23, 210-228, (1991) · Zbl 0722.60096
[6] Deng, Y., On the comparison of point processes, Journal of applied probability, 22, 300-313, (1985) · Zbl 0566.60047
[7] Franken, P.; Kirstein, M., Zur vergleichbarkeit zufälliger prozesse, Mathematische nachrichten, 78, 197-205, (1977) · Zbl 0373.60130
[8] Grandell, J., Aspects of risk theory, (1991), Springer New York/ Berlin/Heidelberg · Zbl 0717.62100
[9] Reinhard, J., On a class of semi-Markov risk models obtained as classical risk models in a Markovian environment, ASTIN bulletin, XIV, 23-43, (1984)
[10] Rolski, T., Queues with nonstationary inputs, Queueing systems, 5, 113-130, (1989) · Zbl 0687.60082
[11] Shaked, M.; Shanthikumar, J., Stochastic orders and their applications, (1994), Academic Press New York · Zbl 0806.62009
[12] Stoyan, D., Comparison methods for queues and other stochastic models, (1983), Wiley Chichester
[13] Szekli, R.; Desney, R.L.; Hur, S., MR/GI/1 queues with positively correlated arrival stream, Journal of applied probability, 31, 497-514, (1994) · Zbl 0802.60088
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.