Title: On the Efficient Simulation of the Left-Tail of the Sum of Correlated Log-normal Variates,
Abstract: The sum of Log-normal variates is encountered in many challenging
applications such as in performance analysis of wireless communication systems
and in financial engineering. Several approximation methods have been developed
in the literature, the accuracy of which is not ensured in the tail regions.
These regions are of primordial interest wherein small probability values have
to be evaluated with high precision. Variance reduction techniques are known to
yield accurate, yet efficient, estimates of small probability values. Most of
the existing approaches, however, have considered the problem of estimating the
right-tail of the sum of Log-normal random variables (RVS). In the present
work, we consider instead the estimation of the left-tail of the sum of
correlated Log-normal variates with Gaussian copula under a mild assumption on
the covariance matrix. We propose an estimator combining an existing
mean-shifting importance sampling approach with a control variate technique.
The main result is that the proposed estimator has an asymptotically vanishing
relative error which represents a major finding in the context of the left-tail
simulation of the sum of Log-normal RVs. Finally, we assess by various
simulation results the performances of the proposed estimator compared to
existing estimators.