Aging properties and bounds for ruin probabilities and stop-loss premiums

In this paper, we derive relations between stop-loss premiums and their associated ruin probabilities by use of the new worse than used (NWU) aging property of the ruin probability. General upper and lower bounds for the stop-loss premium are derived. Also, we get a general upper bound for the ruin probability, the general upper bound is sharper than that of Willmot [Refinements and distributional generalizations of Lundbergʹs inequalities, Insurance: Mathematics and Economics 15 (1994) 49–63] and asymptotically sharper than that of Broeckx et al. [Ordering of risks and ruin probabilities, Insurance: Mathematics and Economics 5 (1986) 35–40]. The asymptotical behavior of these bounds is discussed. The relationships among sub-exponential, new better than used (NBU) distributions and upper bounds of the ruin probability and stop-loss premiums are considered.