Inequality extensions of Prabhu's formula in ruin theory

Prabhu's formula in the classical actuarial risk model expresses the non-ru in probability in the bounded time interval [0, t] as a function of the dis tribution of the total claim amount in that interval. It is valid in case o f an initial risk reserve u = 0 only. We show that simple modifications of Prabhu's expression furnish lower and upper bounds for the ruin probability in case u > 0. These bounds allow to construct good approximations in the safe sense of finite-time ruin probabilities homogeneous extensions of the classical risk model. These estimates may be interesting in practice becaus e no general algorithms are yet available for the numerical exact evaluatio n of finite-time ruin probabilities in such models. (C) 1999 Elsevier Scien ce B.V. All rights reserved.