An Insurance Risk Model with Parisian Implementation Delays

Abstract

Abstract. Inspired by Parisian barrier options in finance (see e.g. Chesney et al. (1997)), a new definition of the event ruin for an insurance risk model is considered. As in Dassios and Wu (2009), the surplus process is allowed to spend time under a pre-specified default level before ruin is recognized. In this paper, we capitalize on the idea of Erlangian horizons (see Asmussen et al. (2002) and Kyprianou and Pistorius (2003)) and, thus assume an implementation delay of a mixed Erlang nature. Using the modern language of scale functions, we study the Laplace transform of this Parisian time to default in an insurance risk model driven by a spectrally negative Lévy process of bounded variation. In the process, a generalization of the two-sided exit problem for this class of processes is further obtained.

Dividend Problem with Parisian Delay for a Spectrally Negative Lévy Risk Process

“…The dividend is paid at the end of this period. A similar problem for a spectrally negative Lévy process of bounded variation was analyzed in [20]. In this paper, we generalize this result for the general spectrally negative Lévy risk process.…”

“…The value function corresponding to the barrier strategy π a,ζ is given by (20). The optimal barrier a * satisfies:…”

“…Quantities (40) and (41) allow calculating numerically the value function v a,ζ (x) given in (20) for the dividend strategy π a,ζ , where the V (q) (x) is defined by (19). Note also that for a general claim size phase-type distribution the barrier strategy is not always optimal.…”

On the Time Spent in the Red by a Refracted Lévy Risk Process

“…In [13], a definition of Parisian ruin is proposed. For this definition of ruin, each excursion of the surplus process U below the critical level b is accompanied by an independent copy of an independent (of U ) random variable.…”

Ruin probabilities in the Cramér–Lundberg model with temporarily negative capital

“…Further, the concept of Parisian ruin has attracted a lot of attention in the literature. Here, the surplus process is allowed to stay negative for a continuous time interval of a fixed or random length, see [8], [7], [15], [14] and for the cumulative situation [13]. In omega models, the insurance company goes bankrupt at a random time at some surplus dependened bankruptcy rate when U t is negative, see [3], [12] and [4].…”