On the ruin probabilities in a general economic environment

Let {A(n) \n = 1, 2,...} and {B-n \n = 1, 2,...} be sequences of random variables and Y-n = B-1 + A(1)B(2) + A(1)A(2)B(3) + ... + A(1) ... A(n-1)B(n). Let M be a positive real number. Define the time of ruin by T-M = inf{n \ Y-n > M} (T-M = +infinity, if Y-n less than or equal to 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 < infinity) approximate to 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 < infinity) similar to CM-w with a strictly positive constant C. (C) 1999 Elsevier Science B.V. All rights reserved.