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 n∈N, denoteYn=B1+A1B2+A1A2B3+⋯+A1⋯An−1Bn+A1⋯AnLn.For M>0, define the time of ruin by TM=inf{n | Yn>M} (TM=+∞, if Yn⩽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⩽x log M)≈M−R(x) and P(TM<∞)≈M−w where x>0 is fixed and R(x) and w are positive parameters. We also prove an asymptotic equivalence P(TM<∞)∼CM−w with a strictly positive constant C. Similar results are obtained in an analogous continuous time model.