Simulating level crossing probabilities by importance sampling for non-decreasing compound Poisson processes with bounded jumps and a negative drift (Q2743919)

Let \(Y_t\) be a nondecreasing compound Poisson process with bounded jumps, let \(c\) be a sufficient large constant such that \(Y_t- ct\to-\infty\) as \(t\to\infty\), and let \(T_b\) be the first time at which \(Y_t- ct\) crosses the level \(b> 0\) (\(T_b= \infty\) if \(Y_t- ct< b\) for all \(t>0\)). Importance sampling is used to estimate \(P(T_b<\infty)\) by Monte Carlo simulation. A unique asymptotically efficient simulation law \((b\to\infty)\) in a sense closely related to large deviations is derived within a certain class of simulation laws.