An algebraic operator approach to the analysis of Gerber-Shiu functions

We introduce an algebraic operator framework to study discounted penalty functions in renewal risk models. For inter-arrival and claim size distributions with rational Laplace transform, the usual integral equation is transformed into a boundary value problem, which is solved by symbolic techniques. The factorization of the differential operator can be lifted to the level of boundary value problems, amounting to iteratively solving first-order problems. This leads to an explicit expression for the Gerber-Shiu function in terms of the penalty function.

A direct approach to the discounted penalty function

This paper provides a new and accessible approach to establishing certain results concerning the discounted penalty function. The direct approach consists of two steps. In the first step, closed-form expressions are obtained in the special case in which the claim amount distribution is a combination of exponential distributions. A rational function is useful in this context. For the second step, one observes that the family of combinations of exponential distributions is dense. Hence, it suffices to reformulate the results of the first step to obtain general results. The surplus process has downward and upward jumps, modeled by two independent compound Poisson processes. If the distribution of the upward jumps is exponential, a series of new results can be obtained with ease. Subsequently, certain results of Gerber and Shiu [H. U. Gerber and E. S. W. Shiu, North American Actuarial Journal 2(1): 48–78 (1998)] can be reproduced. The two-step approach is also applied when an independent Wiener process is added to the surplus process. Certain results are related to Zhang et al. [Z. Zhang, H. Yang, and S. Li, Journal of Computational and Applied Mathematics 233: 1773–1 784 (2010)], which uses different methods.

Gastric supply manipulation to modulate ghrelin production and enhance vascularization to the cardia: proof of the concept in a porcine model.

Introduction. Selective embolization of the left-gastric artery (LGA) reduces levels of ghrelin and achieves significant short-term weight loss. However, embolization of the LGA would prevent the performance of bariatric procedures because the high-risk leakage area (gastroesophageal junction [GEJ]) would be devascularized. Aim. To assess an alternative vascular approach to the modulation of ghrelin levels and generate a blood flow manipulation, consequently increasing the vascular supply to the GEJ. Materials and methods. A total of 6 pigs underwent a laparoscopic clipping of the left gastroepiploic artery. Preoperative and postoperative CT angiographies were performed. Ghrelin levels were assessed perioperatively and then once per week for 3 weeks. Reactive oxygen species (ROS; expressed as ROS/mg of dry weight [DW]), mitochondria respiratory rate, and capillary lactates were assessed before and 1 hour after clipping (T0 and T1) and after 3 weeks of survival (T2), on seromuscular biopsies. A celiac trunk angiography was performed at 3 weeks. Results. Mean (±standard deviation) ghrelin levels were significantly reduced 1 hour after clipping (1902 ± 307.8 pg/mL vs 1084 ± 680.0; P = .04) and at 3 weeks (954.5 ± 473.2 pg/mL; P = .01). Mean ROS levels were statistically significantly decreased at the cardia at T2 when compared with T0 (0.018 ± 0.006 mg/DW vs 0.02957 ± 0.0096 mg/DW; P = .01) and T1 (0.0376 ± 0.008mg/DW; P = .007). Capillary lactates were significantly decreased after 3 weeks, and the mitochondria respiratory rate remained constant over time at the cardia and pylorus, showing significant regional differences. Conclusions. Manipulation of the gastric flow targeting the gastroepiploic arcade induces ghrelin reduction. An endovascular approach is currently under evaluation.