Deficit distributions at ruin in a regime-switching Sparre Andersen model

• Autorzy: Rudź M.

Identyfikatory

DOI

10.1515/jaa-2018-0010

• Języki publikacji: EN

Abstrakty

EN

In this paper, we investigate deficit distributions at ruin in a regime-switching Sparre Andersen model. A Markov chain is assumed to switch the amount and/or respective wait time distributions of claims while the insurer can adjust the premiums in response. Special attention is paid to an operator L generated by the risk process. We show that the deficit distributions at ruin during n periods, given the state of the Markov chain at time zero, form a vector of functions, which is the n-th iteration of L on the vector of functions being identically equal to zero. Moreover, in the case of infinite horizon, the deficit distributions at ruin are shown to be a fixed point of L. Upper bounds for the vector of deficit distributions at ruin are also proven.

Słowa kluczowe

EN

regime-switching Sparre Andersen model; risk operators; deficit distributions at ruin; ruin probabilities; NBU

PL

model przełącznikowy Sparre Andersena; prawdopodobieństwo ruiny; NBU

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2018
• Tom: Vol. 24, nr 1
• Strony: 99--107
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

redaktor

Gajek L.

• Institute of Mathematics, Lodz University of Technology, Wólczańska 215, 90-924 Łódź, Poland

autor

Rudź M.

• Institute of Mathematics, Lodz University of Technology, Wólczańska 215, 90-924 Łódź, Poland

Bibliografia

• [1] S. Asmussen, Risk theory in a Markovian environment, Scand. Actuar. J. (1989), no. 2, 69-100.
• [2] N. Bäuerle, Some results about the expected ruin time in Markov-modulated risk models, Insurance Math. Econom. 18 (1996), no. 2, 119-127.
• [3] S. Frühwirth-Schnatter, Finite Mixture and Markov Switching Models, Springer Ser. Statist., Springer, New York, 2006.
• [4] L. Gajek, On the deficit distribution when ruin occurs-discrete time model, Insurance Math. Econom. 36 (2005), no. 1, 13-24.
• [5] L. Gajek and M. Rudź, Sharp approximations of ruin probabilities in the discrete time models, Scand. Actuar. J. (2013), no. 5, 352-382.
• [6] L. Gajek and M. Rudź, A generalization of Gerber’s inequality for ruin probabilities in risk-switching models, Statist. Probab. Lett. 129 (2017), 236-240.
• [7] L. Gajek and M. Rudź, Banach Contraction Principle and ruin probabilities in regime-switching models, Insurance Math. Econom. 80 (2018), 45-53.
• [8] L. Gajek and M. Rudź, Finite-horizon ruin probabilities in a risk-switching Sparre Andersen model, Methodol. Comput. Appl. Probab. (2018), https://doi.org/10.1007/s11009-018-9627-2.
• [9] B. Kim and H.-S. Kim, Moments of claims in a Markovian environment, Insurance Math. Econom. 40 (2007), no. 3, 485-497.
• [10] S. A. Klugman, H. H. Panjer and G. E. Willmot, Loss Models. From Data to Decisions , Wiley Ser. Probab. Stat., John Wiley & Sons, New York, 1998.
• [11] Y. Lu, On the severity of ruin in a Markov-modulated risk model, Scand. Actuar. J. (2006), no. 4, 183-202.
• [12] Y. Lu and S. Li, On the probability of ruin in a Markov-modulated risk model, Insurance Math. Econom. 37 (2005), no. 3, 522-532.
• [13] J. M. Reinhard, On a class of semi-Markov risk models obtained as classical risk models in a Markovian environment, Astin Bull. 14 (1984), no. 1, 23-43.
• [14] E. Sparre Andersen, On the collective theory of risk in case of contagion between the claims, in: Transactions of the 15th International Congress of Actuaries. Vol. II , International Congress of Actuaries, New York (1957), 219-229.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).