On the analysis of ruin-related quantities in the delayed renewal risk model

详细信息    查看全文

• 作者：So-Yeun Kim^a ; ^{s22kim@hotmail.com" class="auth_mail" title="E-mail the corresponding author} ; Gordon E. Willmot^b
• 关键词：Gerber&ndash ; Shiu function ; Delayed renewal risk model ; Time of ruin ; Deficit at ruin ; Maximal aggregate loss ; Stochastic decomposition ; Compound geometric convolution ; Distributional assumption of time until the first claim
• 刊名：Insurance: Mathematics and Economics
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：66
• 期：Complete
• 页码：77-85
• 全文大小：414 K

文摘

This paper first explores the Laplace transform of the time of ruin in the delayed renewal risk model. We show that height="20" width="37" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0167668715001614-si7.gif">${\bar{G}}_{\delta }^d\left(u\right)$, the Laplace transform of the time of ruin in the delayed model, also satisfies a defective renewal equation and use this to study the Cramer–Lundberg asymptotics and bounds of height="20" width="37" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0167668715001614-si7.gif">${\bar{G}}_{\delta }^d\left(u\right)$. Next, the paper considers the stochastic decomposition of the residual lifetime of maximal aggregate loss and more generally height="20" width="15" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0167668715001614-si9.gif">$L_{\delta }^d$ in the delayed renewal risk model, using the framework equation introduced in Kim and Willmot (2011) and the defective renewal equation for the Laplace transform of the time of ruin. As a result of the decomposition, we propose a way to calculate the mean and the moments of the proper deficit in the delayed renewal risk model. Lastly, closed form expressions are derived for the Gerber–Shiu function in the delayed renewal risk model with the distributional assumption of time until the first claim and simulation results are included to assess the impact of different distributional assumptions on the time until the first claim.