Ruin problems with compounding assets

We consider a generalization of the classical model of collective risk theory. It is assumed that the cumulative income of a firm is given by a process X with stationary independent increments, and that interest is earned continuously on the firm's assets. Then Y(t), the assets of the firm at time t, can be represented by a simple path-wise integral with respect to the income process X. A general characterization is obtained for the probability r(y) that assets will ever fall to zero when the initial asset level is y (the probability of ruin). From this we obtain a general upper bound for r(y), a general solution for the case where X has no negative jumps, and explicit formulas for three particular examples. In addition, an approximation theorem is proved using the weak convergence theory for stochastic processes. This shows that if the income process is well approximated by Brownian motion with drift, then the assets process Y is well approximated by a certain diffusion process Y*, and r(y) is well approximated by a corresponding first passage probability r*(y). The diffusion Y*, which we call compounding Brownian motion, is closely related to the classical Ornstein-Uhlenbeck process.