EXACT AND ASYMPTOTIC SOLUTIONS FOR THE TIME-DEPENDENT PROBLEM OF COLLECTIVE RUIN .1.

The time-dependent problem of collective ruin concerns the fate of a r isk business such as an insurance company. The classic model is given in terms of a risk reserve, Z(t), that increases according to a determ inistic process (premiums) and decreases according to a compound Poiss on process (claims). An integrodifferential backward Kolmogorov equati on is derived for the probability of surviving through time t. Exact s olutions for an exponential claims distribution are derived when the p remiums are modeled as either constant over time, or linearly dependen t on the size of the reserve. These complicated exact solutions are an alyzed asymptotically for three fundamental regions in parameter space and for various regions in the (z,t) plane, where z = Z(0). This work seems to be the first which considers the time-dependent problem with nonconstant premiums.