In the insurance business risky investments are dangerous

We find an exact asymptotics of the ruin probability Psi(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 sigma > 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 rho := 2a/sigma(2) > 1, that Psi(u) similar to Ku(1-rho) for some K > 0. If rho < 1, then Psi(u) = 1.