Ruin probability in Sparre Andersen risk model with claim inter-arrival times distributed as Erlang

Abstract

This article deals with the ruin probability in a Sparre Andersen risk process with the inter-claim times being Erlang distributed in the framework of piecewise deterministic Markov process (PDMP). We construct an exponential martingale by virtue of the extended generator of the PDMP to change the measure. Some results are derived for the ruin probabilities, such as the general expressions for ruin probability, Lundberg bounds, Cramér-Lundberg approximations, and finite-horizon ruin probability.

Keywords

Sparre Andersen risk model / Erlang inter-claim times / ruin probability / Lundberg bound / Cramér-Lundberg approximation

References

[1] Asmussen S. Ruin Probabilities. Singapore: World Scientific, 2000

[2] Cheng Y, Tang Q. Moments of the surplus before ruin and deficit at ruin in the Erlang(2) risk process. N Am Actuar J, 2003, 7: 1−12

[3] Dickson D C M. On a class of renewal risk process. N Am Actuar J, 1998, 2(3): 60−73

[4] Dickson D C M, Hipp C. Ruin probabilities for Erlang(2) risk process. Insurance Math Econom, 1998, 22(3): 251−262

[5] Dickson D C M, Hipp C. On the time to ruin for Erlang(2) risk process. Insurance Math Econom, 2001, 29(3): 333−344

[6] Gao H, Yin C. A perturbed risk process compounded by a geometric Brownian motion with a dividend barrier strategy. Appl Math Comput, 2008, 205(1): 454−464

[7] Li S, Dickson D C M, The maximum surplus before ruin in an Erlang(n) risk process and related problems. Insurance Math Econom, 2006, 38(3): 529−539

[8] Li S, Garrido J. On ruin for the Erlang(n) risk process. Insurance Math Econom, 2004, 34(3): 391−408

[9] Rolski T, Schmidli H, Schimidt V, Teugels J, Stochastic Processes for Insurance and Finance. New York: Jhon Wiley & Sons, 1999

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg