Ruin probability in the presence of risky investments

We consider an insurance company in the case when the premium rate is a bounded non-negative random function $c_{\text{zs}}t$ 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 $\beta :=2a\mathrm{/}{\sigma }^2-1>0$ we find exact the asymptotic upper and lower bounds for the ruin probability $\mathrm{\Psi }(u)$ as the initial endowment $u$ tends to infinity, i.e. we show that $C_{\ast }{u}^{-\beta }\le \mathrm{\Psi }(u)\le C^{\ast }{u}^{-\beta }$ for sufficiently large $u$. Moreover if $c_{\text{zs}}t=c^{\ast }{e}^{\gamma t}$ with $\gamma\le 0$wefindtheexactasymptoticsoftheruinprobability,namely$\Psi(u)\sim u^{-\beta}$\mathrm{.}If$\beta\le 0$,weshowthat$\Psi(u)=1$forany$u\ge 0$.