Lundberg-type bounds for the joint distribution of surplus immediately before and at ruin under the Sparre Andersen model

In this paper we consider the Sparre Andersen insurance risk model. Three cases are discussed: the ordinary renewal risk process, stationary renewal risk process, and s-delayed renewal risk process. In the first part of the paper we study the joint distribution of surplus immediately before and at ruin under the renewal insurance risk model. By constructing an exponential martingale, we obtain Lundberg-type upper bounds for the joint distribution. Consequently we obtain bounds for the distribution of the deficit at ruin and ruin probability. In the second part of the paper, we consider the special case of phase-type claims and rederive the closed-form expression for the distribution of the severity of ruin, obtained by Drekic et al. (2003, 2004). Finally, we present some numerical results to illustrate the tightness of the bounds obtained in this paper.