On the Decomposition of the Ruin Probability for a JumpDiffusion Surplus Process Compounded by a Geometric Brownian Motion

Assume that the surplus of an insurer follows a jump­diffusion process and the insurer would invest its surplus in a risky asset, whose prices are modelled by a geometric Brow­ nian motion. The resulting surplus for the insurer is called as a jump­diffusion surplus process compounded by the geometric Brownian motion. In this resulting surplus process, ruin may be caused by a claim or by oscillation. We decompose the ruin probability in the resulting surplus process into the sum of two ruin probabilities, one is the probability that ruin is caused by a claim and the other is the probability that ruin is caused by oscillation. Integro­differential equations for these ruin probabilities are derived. When claim sizes are exponentially distributed, asymptotical formulas of the ruin probabilities are derived from the integro­differential equations, and it is shown that all the three ruin probabilities are asymptotical power functions with the same orders and that the orders of the power functions are determined by the drift and volatility parameters of the geometric Brownian motion. It is known that the ruin probability for a jump­diffusion surplus process is an exponential function when claim sizes are exponentially distributed. The results of this paper further confirm that risky investments for an insurer are dangerous in the sense that either ruin is certain or the ruin probabilities are asymptotical power functions, not exponential functions, when claim sizes are exponentially distributed.