On Ruin Probabilities in a Sparre Andersen Type Model in the Presence of Risky Investments and Random Switching

We study the asymptotic behavior of ruin probabilities, as the initial reserve goes to infinity, for a reserve process model where claims arrive according to a renewal process, while between the claim times the process has the dynamics of geometric Brownian motion-type Itô processes with time-dependent random coefficients. These coefficients are “reset” after each claim time, switching to new values independent of the past history of the process. We use the implicit renewal theory to obtain power-function bounds for the eventual ruin probability. In the special case when the random drift and diffusion coefficients of the investment returns process remain unchanged between consecutive claim arrivals, we obtain conditions for the validity of the power function decay behaviour (as the initial reserve tends to infinity) for the ruin probability for our model.