Ruin probability in the presence of risky investments

Serguei Pergamenchtchikov (1) , Zeitouny Omar (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem

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 $u\ge 0$.

Mots clés: Risk process; Geometric Brownian motion; Ruin probability

Domaines: Probabilités [math.PR]

Serguei Pergamenchtchikov, Zeitouny Omar. Ruin probability in the presence of risky investments. 2008. ⟨hal-00532597⟩