Finite and infinite time ruin probabilities in a stochastic economic environment

Let (A1,B1,L1),(A2,B2,L2),... be a sequence of independent and identically distributed random vectors. For , denoteYn=B1+A1B2+A1A2B3+...+A1...An-1Bn+A1...AnLn.For M>0, define the time of ruin by TM=inf{n Yn>M} (TM=+[infinity], if Yn[less-than-or-equals, slant]M 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(TM[less-than-or-equals, slant]x log M)[approximate]M-R(x) and P(TM<[infinity])[approximate]M-w where x>0 is fixed and R(x) and w are positive parameters. We also prove an asymptotic equivalence P(TM<[infinity])~CM-w with a strictly positive constant C. Similar results are obtained in an analogous continuous time model.