Ruin probability in the presence of risky investments

Serguei Pergamenchtchikov and Zeitouny Omar
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Serguei Pergamenchtchikov: LMRS
Zeitouny Omar: LMRS

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Abstract: We consider an insurance company in the case when the premium rate is a bounded non-negative random function $c_\zs{t}$ and the capital of the insurance company is invested in a risky asset whose price follows a geometric Brownian motion with mean return $a$ and volatility $\sigma>0$. If $\beta:=2a/\sigma^2-1>0$ we find exact the asymptotic upper and lower bounds for the ruin probability $\Psi(u)$ as the initial endowment $u$ tends to infinity, i.e. we show that $C_*u^{-\beta}\le\Psi(u)\le C^*u^{-\beta}$ for sufficiently large $u$. Moreover if $c_\zs{t}=c^*e^{\gamma t}$ with $\gamma\le 0$ we find the exact asymptotics of the ruin probability, namely $\Psi(u)\sim u^{-\beta}$. If $\beta\le 0$, we show that $\Psi(u)=1$ for any $u\ge 0$.

Date: 2010-11
New Economics Papers: this item is included in nep-ias
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