turkmath.org

We are interested in the asymptotic of the ruin probability for a process describing the evolution of the capital reserve of an insurance company investing its capital reserve in a risky asset with the price given by a geometric Lévy process, e.g. a geometric Brownian motion. Mathematically, the dynamics of reserve can be described by a generalized Ornstein–Uhlenbeck process. To the moment there are two methods of study: based on integro-differential equations for ruin probabilities and the implicit renewal theory. As an example for the company selling annuities, the business process has upward jumps. For the investments, we suppose that the cumulant-generating function H(q) = ln E e^-qV_1 of the increment of log price process V admits a root beta > 0 at which H is continuous while the business activity process is not a subordinator. We show that the ruin probability has the exact asymptotic Cu^−beta as u->infty. We also discuss conditions under which the ruin happens with probability one.