Finite and infinite time ruin probabilities in a stochastic economic environment

Let (A(1), B-1, L-1),(A(2), B-2, L-2),... be a sequence of independent and identically distributed random vectors. For n is an element ofN, denote Y-n=B-1+A(1)B(2)+A(1)A(2)B(3)+...+A(1.)..A(n-1)B(n)+A(1)A(n)L(n). For M > 0, define the time of ruin by T-M=inf{n\Y-n > M} (T-M=+infinity, if Y(n)less than or equal toM for n=1, 2,...). We are interested in the ruin probabilities for large M. Our objective is to give reasons for the crude estimates P(T(M)less than or equal to xlogM) approximate to M-R(x) and P(T-M<<infinity>)approximate toM(-w) where x>0 is fixed and R(x) and w are positive parameters. We also prove an asymptotic equivalence P(T-M<<infinity>)similar to CM-w with a strictly positive constant C. Similar results are obtained in an analogous continuous time model. (C) 2001 Elsevier Science B.V. All rights reserved.