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 c t and the price of the invested risk asset follows a geometric Brownian motion with drift a and volatility σ > 0 . It is proved by Pergamenshchikov and Zeitouny that the probability of ruin, ψ ( u ) , is equal to 1 , for any initial endowment u ≥ 0 , if ρ ≔ 2 a / σ 2 ≤ 1 and the distribution of claim size has an unbounded support. In this paper, we prove that ψ ( u ) = 1 if ρ ≤ 1 without any assumption on the positive claim size.