Inequality extensions of Prabhu’s formula in ruin theory

Prabhu’s formula in the classical actuarial risk model expresses the non-ruin probability in the bounded time interval [0,t] as a function of the distribution of the total claim amount in that interval. It is valid in case of 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 because no general algorithms are yet available for the numerical exact evaluation of finite-time ruin probabilities in such models.