Insurance: Mathematics and Economics 46 (2010) 12–18 Contents lists available at ScienceDirect Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime Finite time ruin problems for the Erlang(2) risk model David C.M. Dickson * , Shuanming Li Centre for Actuarial Studies, Department of Economics, University of Melbourne, Victoria 3010, Australia article info Article history: Received August 2008 Received in revised form March 2009 Accepted 6 May 2009 abstract We consider the Erlang(2) risk model and derive expressions for the density of the time to ruin and the joint density of the time to ruin and the deficit at ruin when the individual claim amount distribution is (i) an exponential distribution and (ii) an Erlang(2) distribution. We also consider the special case when the initial surplus is zero. © 2009 Elsevier B.V. All rights reserved. 1. Introduction In this paper we use results given by Dickson and Hipp (2001) and ideas given in Cheung et al. (2008) and Dickson (2008) to study finite time ruin problems for the Erlang(2) risk model. In particular we aim to find formulae for finite time ruin probabilities and for joint densities of the time to ruin and deficit at ruin. In the literature there are very few exact formulae for finite time ruin probabilities. For the classical risk model, Dickson and Willmot (2005) give a formula for the finite time ruin probability in the case when the individual claims have a distribution that is an infinite mixture of Erlang distributions. Willmot and Woo (2007) explain why this formula covers a range of individual claim amount distributions and covers all cases for which formulae for the finite time ruin probability exist. See Drekic and Willmot (2003) and Garcia (2005). In the case of Sparre Andersen risk models formulae for finite time ruin probabilities exist only in the case of exponential claims — see Dickson et al. (2005) and Borovkov and Dickson (2008). Dickson (2008) found formulae for the joint density of the time to ruin and the deficit at ruin in the classical risk model when the distribution of individual claims was either Erlang(2) or a mixture of two exponential distributions. However, no such results presently exist for Sparre Andersen risk models. Here we apply the methodology in Dickson (2008) to derive such results for the Erlang(2) risk model. The outline of this paper is as follows. In Section 2 we set out the mathematical preliminaries and give transform relationships that are central to our derivations in subsequent sections. In Section 3 we discuss the special case when u = 0 and find that the deri- vation of exact results is somewhat complicated in general, but less so when the individual claim amount distribution has a * Corresponding author. E-mail addresses: dcmd@unimelb.edu.au (D.C.M. Dickson), shli@unimelb.edu.au (S. Li). particular form. The cases of individual claim amounts having (i) an exponential distribution and (ii) an Erlang(2) distribution are discussed in Sections 4 and 5 respectively. We make some concluding remarks in Section 6. 2. Preliminaries We adopt the model of Dickson and Hipp (2001). Thus, we are dealing with a Sparre Andersen risk model under which claim inter-arrival times are distributed as Erlang(2) with scale param- eter β . We denote by p the density function of individual claim amounts, and denote the kth moment as m k . Further, the Laplace transform of p is denoted by ˜ p where ˜ p(s) =  ∞ 0 e -sx p(x)dx. Let P = 1 - ¯ P denote the distribution function. We denote by c the insurer’s premium income per unit time and assume that 2c /β > m 1 . Let {U (t )} t ≥0 denote the surplus process, let T denote the time of ruin, and let Y =|- U (T )| denote the deficit at ruin. Define φ(u) = E  e -δT -sY I (T < ∞)|U (0) = u  to be the bivariate Laplace transform of T and Y . Let w(u, t ) denote the (defective) density of T , and let w(u, y, t ) denote the (defective) joint density of T and Y so that φ(u) =  ∞ 0  ∞ 0 e -δt -sy w(u, y, t )dydt . For this risk model, Lundberg’s fundamental equation is s 2 - 2 β + δ c s +  β + δ c  2 = β 2 c 2 ˜ p(s) and Dickson and Hipp (2001) show that this equation has two solutions r 1 and r 2 such that r 1 < (β + δ)/c < r 2 . We easily deduce 0167-6687/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2009.05.001