State-dependent Importance Sampling for Regularly Varying Random Walks (with J. C. Liu).

Summary:

A classical problem in large deviations analysis for light tailed systems (maybe the most basic one) involves large deviations for the empirical mean of light-tailed random variables. A tutorial on rare-event simulation may well start precisely with this problem, showing how exponential tilting and large deviations ideas give an asymptotically optimal importance sampling estimator. Well, how to design such an optimal estimator for heavy-tailed (regularly varying) random variables is precisely the topic of this paper. We actually can show a stronger optimality notion than what is proved in light-tailed environments and we do it for problems that involve more than one jump for the rare event to occur; for instance, the pricing of an out-of-the-money digital call-in option with heavy-tailed return rates.

Bibtex:

@Article{BlaLiuRW2008,

author = { J. Blanchet and J. C. Liu},

title = {State-dependent importance sampling for regularly varying random walks},

journal = {Advances in Applied Probability},

year = {2008},

volume = {40},

pages = {1104-1128}

}