Parisian ruin of the Brownian motion risk model with constant force of interest

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• 作者：Long Bai^a ; ^{long.bai@unil.ch" class="auth_mail" title="E-mail the corresponding author} ; Li Luo^a ; ^b ; ^{Li.Luo@unil.ch" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 60G15 ; secondary ; 60G70
• 刊名：Statistics & Probability Letters
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：120
• 期：Complete
• 页码：34-44
• 全文大小：424 K

文摘

Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301754&_mathId=si1.gif&_user=111111111&_pii=S0167715216301754&_rdoc=1&_issn=01677152&md5=dc2ff1c94f440f6c68368bed6ad75aaa" title="Click to view the MathML source">B(t),t∈Rclass="mathContainer hidden">class="mathCode">$B\left(t\right),t\in \mathbb{R}$ be a standard Brownian motion. Define a risk process where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301754&_mathId=si3.gif&_user=111111111&_pii=S0167715216301754&_rdoc=1&_issn=01677152&md5=9b26c89c17190855a7af43e01a8b3a01" title="Click to view the MathML source">u≥0class="mathContainer hidden">class="mathCode">$u\ge 0$ is the initial reserve, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301754&_mathId=si4.gif&_user=111111111&_pii=S0167715216301754&_rdoc=1&_issn=01677152&md5=03036eb638c38699982aa87232aa2504" title="Click to view the MathML source">δ≥0class="mathContainer hidden">class="mathCode">$\delta \ge 0$ is the force of interest, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301754&_mathId=si5.gif&_user=111111111&_pii=S0167715216301754&_rdoc=1&_issn=01677152&md5=fd08bda9502e61ecb6a40b21e0f0391f" title="Click to view the MathML source">c>0class="mathContainer hidden">class="mathCode">$c>0$ is the rate of premium and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301754&_mathId=si6.gif&_user=111111111&_pii=S0167715216301754&_rdoc=1&_issn=01677152&md5=085375abd6f15a032bc373e3535f348c" title="Click to view the MathML source">σ>0class="mathContainer hidden">class="mathCode">$\sigma >0$ is a volatility factor. In this contribution we obtain an approximation of the Parisian ruin probability as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301754&_mathId=si8.gif&_user=111111111&_pii=S0167715216301754&_rdoc=1&_issn=01677152&md5=6955b01cd949809e719408029f2f4139" title="Click to view the MathML source">u→∞class="mathContainer hidden">class="mathCode">$u\to \infty$ where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301754&_mathId=si9.gif&_user=111111111&_pii=S0167715216301754&_rdoc=1&_issn=01677152&md5=221ca8a21077e448b6e95e99902237dc" title="Click to view the MathML source">T_uclass="mathContainer hidden">class="mathCode">$T_u$ is a bounded function. Further, we show that the Parisian ruin time of this risk process can be approximated by an exponential random variable. Our results are new even for the classical ruin probability and ruin time which correspond to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216301754&_mathId=si10.gif&_user=111111111&_pii=S0167715216301754&_rdoc=1&_issn=01677152&md5=02e5862a3b4bed322a5d5503117b781a" title="Click to view the MathML source">T_u≡0class="mathContainer hidden">class="mathCode">$T_u\equiv 0$ in the Parisian setting.