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 pan id="mmlsi1" class="mathmlsrc">pan 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∈Rpan>pan class="mathContainer hidden">pan class="mathCode">$B\left(t\right),t\in \mathbb{R}$pan>pan>pan> be a standard Brownian motion. Define a risk process where pan id="mmlsi3" class="mathmlsrc">pan 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≥0pan>pan class="mathContainer hidden">pan class="mathCode">$u\ge 0$pan>pan>pan> is the initial reserve, pan id="mmlsi4" class="mathmlsrc">pan 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≥0pan>pan class="mathContainer hidden">pan class="mathCode">$\delta \ge 0$pan>pan>pan> is the force of interest, pan id="mmlsi5" class="mathmlsrc">pan 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>0pan>pan class="mathContainer hidden">pan class="mathCode">$c>0$pan>pan>pan> is the rate of premium and pan id="mmlsi6" class="mathmlsrc">pan 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>0pan>pan class="mathContainer hidden">pan class="mathCode">$\sigma >0$pan>pan>pan> is a volatility factor. In this contribution we obtain an approximation of the Parisian ruin probability as pan id="mmlsi8" class="mathmlsrc">pan 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→∞pan>pan class="mathContainer hidden">pan class="mathCode">$u\to \infty$pan>pan>pan> where pan id="mmlsi9" class="mathmlsrc">pan 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_upan>pan class="mathContainer hidden">pan class="mathCode">$T_u$pan>pan>pan> 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 pan id="mmlsi10" class="mathmlsrc">pan 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≡0pan>pan class="mathContainer hidden">pan class="mathCode">$T_u\equiv 0$pan>pan>pan> in the Parisian setting.