In the insurance business risky investments are dangerous: the case of negative risk sums

We find an exact asymptotics of the ruin probability Ψ(u) when the capital of insurance company is invested in a risky asset whose price follows a geometric Brownian motion with mean return a and volatility σ > 0. In contrast to the classical case of non-risky investments where the ruin probability decays exponentially as the initial endowment u tends to infinity, in this model we have, if ρ := 2a/σ > 1, that Ψ(u) ∼ Ku1−ρ for some K > 0. If ρ < 1, then Ψ(u) = 1.