Discounted penalty function at Parisian ruin for Lévy insurance risk process

Abstract

In the setting of a Lévy insurance risk process, we present some results regarding the Parisian ruin problem which concerns the occurrence of an excursion below zero of duration bigger than a given threshold r. First, we give the joint Laplace transform of ruin-time and ruin-position (possibly killed at the first-passage time above a fixed level b), which generalises known results concerning Parisian ruin. This identity can be used to compute the expected discounted penalty function via Laplace inversion. Second, we obtain the q-potential measure of the process killed at Parisian ruin. The results have semi-explicit expressions in terms of the q-scale function and the distribution of the Lévy process.

Draw-down Parisian ruin for spectrally negative Lévy processes

“…The following result presents the joint Laplace transform involving the draw-down Parisian ruin time, the position of X at the draw-down Parisian ruin time, and its running supremum until the draw-down Parisian ruin time. It generalizes Theorem 3.1 in [22]. × e λξ (w)…”

“…More recently, in [22] a novel approach is adopted by connecting the desired Parisian ruin fluctuation quantity with the solution to the Kolmogorov forward equation for a spectrally negative Lévy process to find the joint Laplace transform of the Parisian ruin time and the Parisian ruin position, as well as an expression for the q-potential measure of the process killed at the Parisian ruin time.…”

Parisian Excursion Below a Fixed Level from the Last Record Maximum of Lévy Insurance Risk Process

“…The proof is established for the case where X has paths of bounded and unbounded variation. To deal with unbounded variation case, we will use a limiting argument similar to the one employed in [17], [16] and adjust the ruin time (1.7) accordingly. For this reason, we introduce for ǫ ≥ 0 the stopping time τ ǫ r defined by τ ǫ r = inf{t > r : t − g ǫ t ≥ r} with g ǫ t := sup{s < t : Y s ≤ a − ǫ}.…”

Parisian types of ruin probabilities for a class of dependent risk-reserve processes

“…To avoid this complication, we employ the following standard trick (see, e.g. Loeffen et al 2013or Baurdoux et al 2016).…”

“…The concept of Parisian ruin with deterministic clocks was first studied in Dassios & Wu (2008) for a Cramér-Lundberg process with exponential claim sizes. The results were later generalized to the case of spectrally negative Lévy processes in Czarna & Palmowski (2011) and Loeffen et al (2013) for the case of deterministic clocks, and in Landriault et al (2014) and Baurdoux et al (2016) for the case of random clocks, Erlang and exponential, respectively. In Czarna et al (2017), the distribution of the number of claims leading to Parisian ruin is computed for the Cramér-Lundberg process.…”