On the ruin probabilities in a general economic environment

Abstract

Let {A n | n=1,2,…} and {B n | n=1,2,…} be sequences of random variables and Y n=B 1+A 1B 2+A 1A 2B 3+⋯+A 1⋯A n−1B n. Let M be a positive real number. Define the time of ruin by T M= inf{n | Y n>M} (T M=+∞ , if Y n ⩽ M for n=1,2,…). We are interested in the ruin probabilities for large M. We assume that the sequences { A n } and { B n } are independent and that the variables A 1, A 2,… are strictly positive. The sequences are allowed to be general in other respects. Our main objective is to give reasons for the crude estimate P(T M<∞)≈M −w where w is a positive parameter. In the particular case where both { A n } and { B n } are sequences of independent and identically distributed random variables, we prove an asymptotic equivalence P(T M<∞)∼CM −w with a strictly positive constant C.