Risk Processes with Compounding Assets.
Abstract
The paper considers a generalization of the classical model of collective risk theory. It is assumed that the cumulative income of a firm is given by a stochastic process X(t) with stationary, independent increments, and that interest is earned continuously on the firm's cash assets. It is shown that Y(t), the assets of the firm at time t, can be expressed as a simple path-wise integral with respect to the income process X(.). The behavior of the assets process is studied, much of the analysis focusing on the probability r(y) that assets will ever fall to zero when the initial asset level is y. The author calls r(.) the ruin function. A variety of general results are proved, culminating in a useful characterization of the ruin function. From this the author obtains a general upper bound for r(y) and a general solution for r(y) in the case where X(.) has no negative jumps. The ruin function is explicitly calculated for three particular examples. In addition, an approximation theorem is proved using the formal machinery of the weak convergence theory for stochastic processes.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 22, 1975
- Accession Number
- ADA012984
Entities
People
- J. Michael Harrison
Organizations
- Stanford University