Rare-event simulation for a multidimensional random walk with t distributed increments

We consider the problem of efficient estimation of first passage time probabilities for a multidimensional random walk with t distributed increments, via simulation. In addition of being a natural generalization of the problem of computing ruin probabilities in insurance - in which the focus is a one dimensional random walk - this problem captures important features of large deviations for multidimensional heavy-tailed processes (such as the role played by the mean of the random walk in connection to the spatial location of the target set). We develop a state-dependent importance sampling estimator for this class of multidimensional problems. Then, we argue - using techniques based on Lyapunov type inequalities - that our estimator is strongly efficient.