Ruin probability in the presence of risky investments

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 $C_{\ast }{u}^{-\beta }\le \Psi (u)\le C^{\ast }{u}^{-\beta }$ for sufficiently large $u$. Moreover if $c_\zs{t}=c^*e^{γt}$ 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 $u\ge 0$.