On the stop-loss transform and order for the surplus process perturbed by diffusion

On the limit behavior of iterated equilibrium distributions for the Gamma and Weibull families

“…where the last equality follows by applying Lemma 9. The proof is concluded by rewriting (6) taking into account (12) and (13).…”

“…Remark 4 It follows from (3) that the s−iterated distribution may be interpreted as a normalized survival moment of order s − 1. It is also worth mentioning that (3) means that, up to a normalizing factor, the s−iterated distribution is the stop-loss transform of order s − 1, E(X − x) s−1 + , a common quantity of interest in actuarial models (see, for example, Cheng and Pai [4], Nair et al [8], Tsai [12] or Rachev and Rüschendorf [11], among many other references).…”

An effective method for constructing bounds for ruin probabilities for the surplus process perturbed by diffusion

“…(3.10) is replaced with (see Tsai (2006)) K Y k (u)/ 0u X (/D 1 e (s 1 u 'D 2 e (s 2 u )/; u #[0/; u k ]/; where…”

“…Previously published articles focused on c t,X (u)5/c t,Y (u) or c X (u) 5/c Y (u) for all u]0 under some conditions; see, for example, Daley & Rolski (1984), Cheng & Pai (2003), and Tsai (2006Tsai ( , 2008Tsai ( , 2009). One of these conditions regards some orderings of claim size random variables X and Y, for instance, X5 mrl Y or X5 sl Y which holds in general only if X and Y come from the same distribution family with different parameters (see Example 3 202 in Tsai (2008)).…”

Some Stochastic Orders over an Interval with Applications

“…Cheng and Pai (2003) studied the stop loss ordering of claim severities, and Denuit et al (2005) suggested stochastic bounds on functions of dependent insurance risks. Moreover, Mitric and Trufin (2016) applied stochastic ordering tools in order to study some properties of a ruin measure related to the maximal aggregate loss, while Tsai (2006) and Tsai (2009) used stochastic order concepts to compare ruin probabilities for two surplus processes perturbed by diffusion. Also, Escudero and Ortega (2008) proposed Laplace transform comparison bounds and results for the insurer's aggregate claims assuming dependence among the random retention levels.…”