We consider an insurance company in the case when the premium rate is abounded non-negative random function $c_\zs{t}$ and the capital of theinsurance company is invested in a risky asset whose price follows a geometricBrownian motion with mean return $a$ and volatility $\sigma>0$. If$\beta:=2a/\sigma^2-1>0$ we find exact the asymptotic upper and lower boundsfor the ruin probability $\Psi(u)$ as the initial endowment $u$ tends toinfinity, i.e. we show that $C_*u^{-\beta}\le\Psi(u)\le C^*u^{-\beta}$ forsufficiently large $u$. Moreover if $c_\zs{t}=c^*e^{\gamma t}$ with $\gamma\le0$ we find the exact asymptotics of the ruin probability, namely $\Psi(u)\simu^{-\beta}$. If $\beta\le 0$, we show that $\Psi(u)=1$ for any $u\ge 0$.
展开▼
机译：当保费率超过非负随机函数$ c_ \ zs {t} $且保险公司的资本投资于价格遵循几何布朗运动且均值收益为$ a的风险资产时，我们考虑一家保险公司$和波动率$ \ sigma> 0 $。如果$ \ beta：= 2a / \ sigma ^ 2-1> 0 $，我们可以找到破产概率$ \ Psi（u）$的精确渐进上界和下界，因为初始end赋$ u $趋于无穷大，即表明$ C_ * u ^ {-\ beta} \ le \ Psi（u）\ le C ^ * u ^ {-\ beta} $足够大的$ u $。此外，如果$ c_ \ zs {t} = c ^ * e ^ {\ gamma t} $与$ \ gamma \ le0 $一起，我们会找到破产概率的精确渐近线，即$ \ Psi（u）\ simu ^ {- \ beta} $。如果$ \ beta \ le 0 $，我们表明对于任何$ u \ ge 0 $，$ \ Psi（u）= 1 $。
展开▼
