Efficient Simulation of Tail Probabilities of Dependent Random Variables (with L. Rojas-Nandayapa).

Summary:

We started to think about this problem after Leonardo visited Columbia in 2009. We first came up with a conditional Monte Carlo algorithm in polar coordinates for sums of correlated lognormals (one conditions on the spherical component of the underlying Gaussian and one integrates in closed form the contribution of the contribution of the radial component). We reported the asymptotic optimality properties in a previous paper co-authored with Soren Asmussen and Sandeep Juneja. However, it was quite natural to conjecture that the same optimality would hold in greater generality; on sums of log-elliptical distributions with basically unbounded radial component. We prove this conjecture under mild assumptions on the density of R. We also have another result that allows to do asymptotically optimal importance sampling for sums of dependent random variables whose marginals are suitably heavy-tailed (ruling out Weibullians but including regularly varying and log-normal). The results are applied to several models in finance such as Variance-Gamma processes, NIG (normal-inverse Gaussian) and a multivariate model of Steve Kou.

Bibtex:

@Article{BlaRoj11,

author = {J. Blanchet and L. Rojas-Nandayapa},

title = {Efficient simulation of tail probabilities of dependent random variables},

journal = {To appear in Journal of Applied Probability},

year = {2011},

volume = {},

pages = {}

}