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On the efficient simulation of the left-tail of the sum of correlated log-normal variates

Author

Listed:
  • Alouini Mohamed-Slim

    (Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal23955, Saudi Arabia)

  • Ben Rached Nadhir

    (Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal23955, Saudi Arabia)

  • Kammoun Abla

    (Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal23955, Saudi Arabia)

  • Tempone Raul

    (Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal23955, Saudi Arabia)

Abstract

The sum of log-normal variates is encountered in many challenging applications such as performance analysis of wireless communication systems and financial engineering. Several approximation methods have been reported in the literature. However, these methods are not accurate in the tail regions. These regions are of primordial interest as 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 have focused on estimating the right-tail of the sum of log-normal random variables (RVs). Here, we instead consider 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. This 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 perform simulations to evaluate the performances of the proposed estimator in comparison with existing ones.

Suggested Citation

  • Alouini Mohamed-Slim & Ben Rached Nadhir & Kammoun Abla & Tempone Raul, 2018. "On the efficient simulation of the left-tail of the sum of correlated log-normal variates," Monte Carlo Methods and Applications, De Gruyter, vol. 24(2), pages 101-115, June.
    • Handle: RePEc:bpj:mcmeap:v:24:y:2018:i:2:p:101-115:n:3
    DOI: 10.1515/mcma-2018-0009
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    References listed on IDEAS

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    1. Søren Asmussen & Jens Ledet Jensen & Leonardo Rojas-Nandayapa, 2016. "Exponential Family Techniques for the Lognormal Left Tail," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(3), pages 774-787, September.
    2. Archil Gulisashvili & Peter Tankov, 2013. "Tail behavior of sums and differences of log-normal random variables," Papers 1309.3057, arXiv.org, revised Jan 2016.
    3. Asmussen, Søren & Rojas-Nandayapa, Leonardo, 2008. "Asymptotics of sums of lognormal random variables with Gaussian copula," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2709-2714, November.
    4. Søren Asmussen & José Blanchet & Sandeep Juneja & Leonardo Rojas-Nandayapa, 2011. "Efficient simulation of tail probabilities of sums of correlated lognormals," Annals of Operations Research, Springer, vol. 189(1), pages 5-23, September.
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