Ruin probability in the Cramér–Lundberg model with risky investments

We consider the Cramér–Lundberg model with investments in an asset with large volatility, where the premium rate is a bounded nonnegative random function ctct and the price of the invested risk asset follows a geometric Brownian motion with drift aa and volatility σ>0σ>0. It is proved by Pergamenshchikov and Zeitouny that the probability of ruin, ψ(u)ψ(u), is equal to 11, for any initial endowment u≥0u≥0, if ρ≔2a/σ2≤1ρ≔2a/σ2≤1 and the distribution of claim size has an unbounded support. In this paper, we prove that ψ(u)=1ψ(u)=1 if ρ≤1ρ≤1 without any assumption on the positive claim size.