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Approximating the Laplace transform of the sum of dependent lognormals

Part of: Distribution theory - Probability Probabilistic methods, simulation and stochastic differential equations Integral transforms, operational calculus

Published online by Cambridge University Press:  25 July 2016

Patrick J. Laub ,
Søren Asmussen ,
Jens L. Jensen  and
Leonardo Rojas-Nandayapa
Patrick J. Laub*
Affiliation:
The University of Queensland and Aarhus University
Søren Asmussen*
Affiliation:
Aarhus University
Jens L. Jensen*
Affiliation:
Aarhus University
Leonardo Rojas-Nandayapa*
Affiliation:
The University of Queensland
*
Department of Mathematics, The University of Queensland, Brisbane, Queensland 4072, Australia. Email address: [email protected]
Department of Mathematics, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark. Email address: [email protected]
Department of Mathematics, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark. Email address: [email protected]
Department of Mathematics, The University of Queensland, Brisbane, Queensland 4072, Australia. Email address: [email protected]
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Abstract

Let (X1,...,Xn) be multivariate normal, with mean vector 𝛍 and covariance matrix 𝚺, and let Sn=eX1+⋯+eXn. The Laplace transform ℒ(θ)=𝔼e-θSn∝∫exp{-hθ(𝒙)}d𝒙 is represented as ℒ̃(θ)I(θ), where ℒ̃(θ) is given in closed form and I(θ) is the error factor (≈1). We obtain ℒ̃(θ) by replacing hθ(𝒙) with a second-order Taylor expansion around its minimiser 𝒙*. An algorithm for calculating the asymptotic expansion of 𝒙* is presented, and it is shown that I(θ)→ 1 as θ→∞. A variety of numerical methods for evaluating I(θ) is discussed, including Monte Carlo with importance sampling and quasi-Monte Carlo. Numerical examples (including Laplace-transform inversion for the density of Sn) are also given.

Keywords

Lognormal distributionasymptoticssaddlepoint approximationimportance samplingquasi-Monte Carlonumerical Laplace-transform inversionLambert W function

MSC classification

Primary: 60E10: Characteristic functions; other transforms
Secondary: 44A10: Laplace transform 65C05: Monte Carlo methods
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue A: Probability, Analysis and Number Theory , July 2016 , pp. 203 - 215
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1] Abate, J. and Whitt, W. (2006).A unified framework for numerically inverting Laplace transforms.INFORMS J. Computing 18,408–421.CrossRefGoogle Scholar
[2] Abu-Dayya, A. A. and Beaulieu, N. C. (1994).Outage probabilities in the presence of correlated lognormal interferers.IEEE Trans. Vehicular Technology 43,164–173.CrossRefGoogle Scholar
[3] Aitchison, J. and Brown, J. A. C. (1957).The Lognormal Distribution with Special Reference to its Uses in Economics.Cambridge University Press.Google Scholar
[4] Asmussen, S. and Glynn, P. W. (2007).Stochastic Simulation: Algorithms and Analysis (Stoch. Modelling Appl. Prob. 57).Springer,New York.CrossRefGoogle Scholar
[5] Asmussen, S. and Rojas-Nandayapa, L. (2008).Asymptotics of sums of lognormal random variables with Gaussian copula.Statist. Prob. Lett. 78,2709–2714.CrossRefGoogle Scholar
[6] Asmussen, S.,Jensen, J. L. and Rojas-Nandayapa, L. (2016).On the Laplace transform of the lognormal distribution. To appear in Methodology Comput. Appl. Prob., 18pp.CrossRefGoogle Scholar
[7] Asmussen, S.,Jensen, J. L. and Rojas-Nandayapa, L. (2016).Exponential family techniques in the lognormal left tail. To appear in Scand. J. Statist. CrossRefGoogle Scholar
[8] Avdis, E. and Whitt, W. (2007).Power algorithms for inverting Laplace transforms.INFORMS J. Computing 19,341–355.CrossRefGoogle Scholar
[9] Beaulieu, N. C. and Rajwani, F. (2004).Highly accurate simple closed-form approximations to lognormal sum distributions and densities.IEEE Commun. Lett. 8,709–711.CrossRefGoogle Scholar
[10] Beaulieu, N. C. and Xie, Q. (2004).An optimal lognormal approximation to lognormal sum distributions.IEEE Trans. Vehicular Technology 53,479–489.CrossRefGoogle Scholar
[11] Beaulieu, N. C.,Abu-Dayya, A. A. and McLane, P. J. (1995).Estimating the distribution of a sum of independent lognormal random variables.IEEE Trans. Commun. 43,2869–2873.CrossRefGoogle Scholar
[12] Corless, R. M.,Gonnet, G. H.,Hare, D. E. G.,Jeffrey, D. J. and Knuth, D. E. (1996).On the Lambert W function.Adv. Comput. Math. 5,329–359.CrossRefGoogle Scholar
[13] Crow, E. L. and Shimizu, K. (eds) (1988).Lognormal Distributions: Theory and Applications (Statist. Textb. Monogr. 88).Marcel Dekker,New York.Google Scholar
[14] Duellmann, K. (2010).Regulatory capital. In Encyclopedia of Quantitative Finance, Vol. IV, ed. R. Cont,John Wiley,New York, pp. 1525–1538.Google Scholar
[15] Dufresne, D. (2004).The log-normal approximation in financial and other computations.Adv. Appl. Prob. 36,747–773.CrossRefGoogle Scholar
[16] Dufresne, D. (2009).Sums of lognormals. Tech. Rep., Centre for Actuarial Sciences, University of Melbourne.Google Scholar
[17] Embrechts, P.,Puccetti, G.,Rüschendorf, L.,Wang, R. and Beleraj, A. (2014).An academic response to Basel 3.5.Risks 2,25–48.CrossRefGoogle Scholar
[18] Fenton, L. (1960).The sum of log-normal probability distributions in scatter transmission systems.IRE Trans. Commun. Systems 8,57–67.CrossRefGoogle Scholar
[19] Gao, X.,Xu, H. and Ye, D. (2009).Asymptotic behavior of tail density for sum of correlated lognormal variables.Internat. J. Math. Math. Sci. 2009, 28pp.CrossRefGoogle Scholar
[20] Glasserman, P. (2003).Monte Carlo Methods in Financial Engineering (Stoch. Model. Appl. Prob. 53).Springer,New York.CrossRefGoogle Scholar
[21] Gulisashvili, A. and Tankov, P. (2016).Tail behavior of sums and differences of log-normal random variables. To appear in Bernoulli. Available at http://www.e-publications.org/ims/submission/BEJ/user/submissionFile/17119?confirm=ef609013, 46pp.CrossRefGoogle Scholar
[22] Johnson, N. L.,Kotz, S. and Balakrishnan, N. (1994).Continuous Univariate Distributions, Vol. 1,2nd edn.John Wiley,New York.Google Scholar
[23] Laub, P. J.,Asmussen, S.,Jensen, J. L. and Rojas-Nandayapa, L. (2016).Online accompaniment for ``Approximating the Laplace transform of the sum of dependent lognormals''. Available at https://github.com/Pat-Laub/SLNLaplaceTransformApprox.Google Scholar
[24] Limpert, E.,Stahel, W. A. and Abbt, M. (2001).Log-normal distributions across the sciences: keys and clues.Bioscience 51,341–352.CrossRefGoogle Scholar
[25] Mallet, A. (2000).Numerical inversion of Laplace transform. Mathematica package. Available at http://library.wolfram.com/infocenter/MathSource/2691/.Google Scholar
[26] Markowitz, H. (1952).Portfolio selection.J. Finance 7,77–91.Google Scholar
[27] McNeil, A. J.,Frey, R. and Embrechts, P. (2015).Quantitative Risk Management: Concepts, Techniques and Tools,2nd edn.Princeton University Press.Google Scholar
[28] Milevsky, M. A. and Posner, S. E. (1998).Asian options, the sum of lognormals, and the reciprocal gamma distribution.J. Financial Quant. Anal. 33,409–422.CrossRefGoogle Scholar
[29] Schwartz, S. C. and Yeh, Y.-S. (1982).On the distribution function and moments of power sums with log-normal components.Bell System Tech. J. 61,1441–1462.CrossRefGoogle Scholar
[30] Stehfest, H. (1970).Algorithm 368: Numerical inversion of Laplace transforms [D5].Commun. ACM 13,47–49.CrossRefGoogle Scholar