On the ruin probabilities in a general economic environment

摘要

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-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<=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