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 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 σ > 0 . If β ≔ 2 a / σ 2 - 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 c t = c * e γ t 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 .