Use of differential and integral inequalities to bound ruin and queuing probabilities

Summary Probabilities of ruin (or non-ruin) are solutions of differential or integro-differential equations. These equations contain terms which cause difficulty in obtaining analytical solutions. One partial way out of this difficulty is to omit the awkward terms, or modify them, to produce a more tractable equation. The price to be paid for this is that the equation becomes a differential or integro-differential inequality. The paper concentrates on applying the theory of such inequalities to obtain bounds on probability of ruin over finite and infinite times respectively. By these methods it is possible to 1. (1) sharpen Lundberg's inequality;2. (2) order (in some cases) probabilities of ruin over infinite time in accordance with the properties of different claim size distributions;3. (3) give an upper bound for finite-time probability of non-ruin in terms only of probabilities involving (a) zero initial reserves, and (b) infinite time. Finally, the application of these results to queuing theory is indicated.