Source: J. Appl. Probab. Volume 47, Number 2
(2010), 301-322.
We consider the problem of efficient estimation via simulation of first passage
time probabilities for a multidimensional random walk with heavy-tailed
increments. In addition to being a natural generalization to the problem of
computing ruin probabilities in insurance - in which the focus is the maximum
of a one-dimensional random walk with negative drift - this problem captures
important features of large deviations for multidimensional heavy-tailed
processes (such as the role played by the mean of the process in connection to
the location of the target set). We develop a state-dependent importance
sampling estimator for this class of multidimensional problems. Then, using
techniques based on Lyapunov inequalities, we argue that our estimator is
strongly efficient in the sense that the relative mean squared error of
our estimator can be made arbitrarily small by increasing the number of
replications, uniformly as the probability of interest approaches 0.
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