Ruin Probabilities and Penalty Functions with Stochastic Rates of Interest

Assume that a compound Poisson surplus process is invested in a stochastic interest process which is assumed to be a Lévy process. We derive recursive and integral equations for ruin probabilities with such an investment. Lower and upper bounds for the ultimate ruin probability are obtained from these equations. When the interest process is a Brownian motion with drift, we give a unified treatment to ruin quantities by studying the expected discounted penalty function associated with the time of ruin. An integral equation for the penalty function is given. Smooth properties of the penalty function are discussed based on the integral equation. Errors in a known result about the smooth properties of the ruin probabilities are corrected. Using a differential argument and moments of exponential functionals of Brownian motions, we derive an integro-differential equation satisfied by the penalty function. Applications of the integro-differential equation are given to the Laplace transform of the time of ruin, the deficit at ruin, the amount of claim causing ruin, and so on. Some known results about ruin quantities are recovered from the generalized penalty function.