On the generalization of the hazard rate twisting-based simulation approach

Estimating the probability that a sum of random variables (RVs) exceeds a given threshold is a well-known challenging problem. A naive Monte Carlo simulation is the standard technique for the estimation of this type of probability. However, this approach is computationally expensive, especially when dealing with rare events. An alternative approach is represented by the use of variance reduction techniques, known for their efficiency in requiring less computations for achieving the same accuracy requirement. Most of these methods have thus far been proposed to deal with specific settings under which the RVs belong to particular classes of distributions. In this paper, we propose a generalization of the well-known hazard rate twisting Importance Sampling-based approach that presents the advantage of being logarithmic efficient for arbitrary sums of RVs. The wide scope of applicability of the proposed method is mainly due to our particular way of selecting the twisting parameter. It is worth observing that this interesting feature is rarely satisfied by variance reduction algorithms whose performances were only proven under some restrictive assumptions. It comes along with a good efficiency, illustrated by some selected simulation results comparing the performance of the proposed method with some existing techniques

On the Efficient Simulation of the Left-Tail of the Sum of Correlated Log-normal Variates

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