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">pan>pan>pan> be a standard Brownian motion. Define a risk process
mula" id="fd000005">
pan class="offscreen">equationpan>pan aria-hidden="true">(pan>0.1pan aria-hidden="true">)pan>
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">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">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">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">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">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">Tupan>pan class="mathContainer hidden">pan class="mathCode">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">Tu≡0pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> in the Parisian setting.