Using Simulation to Estimate First Passage Distribution

Consider a discrete time Markov process {X_n, n > 0}. For a given subset \scr{A} of the state space, consider the problem of using simulation to estimate the number of transitions it takes the process to enter \scr{A}. Using estimators based on the "observed hazard," we are able to improve on the usual Monte Carlo estimator. We also consider the problem of estimating the distribution of the first state in, \scr{A} to be reached, and then extend our results to continuous time.