Simulating level-crossing probabilities by importance sampling

Let X1, X2, · ·· be independent and identically distributed random variables such that ΕΧ1 < 0 and P(X1 ≥ 0) ≥ 0. Fix M ≥ 0 and let T = inf {n: X1 + X2 + · ·· + Xn ≥ M} (T = +∞, if for every n = 1,2, ···). In this paper we consider the estimation of the level-crossing probabilities P(T <∞) and , by using Monte Carlo simulation and especially importance sampling techniques. When using importance sampling, precision and efficiency of the estimation depend crucially on the choice of the simulation distribution. For this choice we introduce a new criterion which is of the type of large deviations theory; consequently, the basic large deviations theory is the main mathematical tool of this paper. We allow a wide class of possible simulation distributions and, considering the case that M →∞, we prove asymptotic optimality results for the simulation of the probabilities P(T <∞) and . The paper ends with an example.