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 ct 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 σ>0. If β≔2a/σ2-1>0 we find exact the asymptotic upper and lower bounds for the ruin probability Ψ(u) as the initial endowment u tends to infinity, i.e. we show that C*u-β⩽Ψ(u)⩽C*u-β for sufficiently large u. Moreover if ct=c*eγt with γ⩽0 we find the exact asymptotics of the ruin probability, namely Ψ(u)∼u-β. If β⩽0, we show that Ψ(u)=1 for any u⩾0.