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Approximating the Laplace transform of the sum of dependent lognormals

  • Patrick J. Laub (a1), Søren Asmussen (a2), Jens L. Jensen (a2) and Leonardo Rojas-Nandayapa (a3)
Abstract
Abstract

Let (X 1,...,X n ) be multivariate normal, with mean vector 𝛍 and covariance matrix 𝚺, and let S n =e X 1 +⋯+e X n . The Laplace transform ℒ(θ)=𝔼eS n ∝∫exp{-h θ(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 h θ(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 S n ) are also given.

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Copyright
Corresponding author
Department of Mathematics, The University of Queensland, Brisbane, Queensland 4072, Australia. Email address: p.laub@uq.edu.au
Department of Mathematics, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark. Email address: asmus@imf.au.dk
Department of Mathematics, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark. Email address: jlj@math.au.dk
Department of Mathematics, The University of Queensland, Brisbane, Queensland 4072, Australia. Email address: l.rojasnandayapa@uq.edu.au
References
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[1] AbateJ. and WhittW. (2006).A unified framework for numerically inverting Laplace transforms.INFORMS J. Computing 18,408421.
[2] Abu-DayyaA. A. and BeaulieuN. C. (1994).Outage probabilities in the presence of correlated lognormal interferers.IEEE Trans. Vehicular Technology 43,164173.
[3] AitchisonJ. and BrownJ. A. C. (1957).The Lognormal Distribution with Special Reference to its Uses in Economics.Cambridge University Press.
[4] AsmussenS. and GlynnP. W. (2007).Stochastic Simulation: Algorithms and Analysis (Stoch. Modelling Appl. Prob. 57).Springer,New York.
[5] AsmussenS. and Rojas-NandayapaL. (2008).Asymptotics of sums of lognormal random variables with Gaussian copula.Statist. Prob. Lett. 78,27092714.
[6] AsmussenS.,JensenJ. L. and Rojas-NandayapaL. (2016).On the Laplace transform of the lognormal distribution. To appear in Methodology Comput. Appl. Prob., 18pp.
[7] AsmussenS.,JensenJ. L. and Rojas-NandayapaL. (2016).Exponential family techniques in the lognormal left tail. To appear in Scand. J. Statist.
[8] AvdisE. and WhittW. (2007).Power algorithms for inverting Laplace transforms.INFORMS J. Computing 19,341355.
[9] BeaulieuN. C. and RajwaniF. (2004).Highly accurate simple closed-form approximations to lognormal sum distributions and densities.IEEE Commun. Lett. 8,709711.
[10] BeaulieuN. C. and XieQ. (2004).An optimal lognormal approximation to lognormal sum distributions.IEEE Trans. Vehicular Technology 53,479489.
[11] BeaulieuN. C.,Abu-DayyaA. A. and McLaneP. J. (1995).Estimating the distribution of a sum of independent lognormal random variables.IEEE Trans. Commun. 43,28692873.
[12] CorlessR. M.,GonnetG. H.,HareD. E. G.,JeffreyD. J. and KnuthD. E. (1996).On the Lambert W function.Adv. Comput. Math. 5,329359.
[13] CrowE. L. and ShimizuK. (eds) (1988).Lognormal Distributions: Theory and Applications (Statist. Textb. Monogr. 88).Marcel Dekker,New York.
[14] DuellmannK. (2010).Regulatory capital. In Encyclopedia of Quantitative Finance, Vol. IV, ed. R. Cont,John Wiley,New York, pp. 15251538.
[15] DufresneD. (2004).The log-normal approximation in financial and other computations.Adv. Appl. Prob. 36,747773.
[16] DufresneD. (2009).Sums of lognormals. Tech. Rep., Centre for Actuarial Sciences, University of Melbourne.
[17] EmbrechtsP.,PuccettiG.,RüschendorfL.,WangR. and BelerajA. (2014).An academic response to Basel 3.5.Risks 2,2548.
[18] FentonL. (1960).The sum of log-normal probability distributions in scatter transmission systems.IRE Trans. Commun. Systems 8,5767.
[19] GaoX.,XuH. and YeD. (2009).Asymptotic behavior of tail density for sum of correlated lognormal variables.Internat. J. Math. Math. Sci. 2009, 28pp.
[20] GlassermanP. (2003).Monte Carlo Methods in Financial Engineering (Stoch. Model. Appl. Prob. 53).Springer,New York.
[21] GulisashviliA. and TankovP. (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.
[22] JohnsonN. L.,KotzS. and BalakrishnanN. (1994).Continuous Univariate Distributions, Vol. 1,2nd edn.John Wiley,New York.
[23] LaubP. J.,AsmussenS.,JensenJ. L. and Rojas-NandayapaL. (2016).Online accompaniment for ``Approximating the Laplace transform of the sum of dependent lognormals''. Available at https://github.com/Pat-Laub/SLNLaplaceTransformApprox.
[24] LimpertE.,StahelW. A. and AbbtM. (2001).Log-normal distributions across the sciences: keys and clues.Bioscience 51,341352.
[25] MalletA. (2000).Numerical inversion of Laplace transform. Mathematica package. Available at http://library.wolfram.com/infocenter/MathSource/2691/.
[26] MarkowitzH. (1952).Portfolio selection.J. Finance 7,7791.
[27] McNeilA. J.,FreyR. and EmbrechtsP. (2015).Quantitative Risk Management: Concepts, Techniques and Tools,2nd edn.Princeton University Press.
[28] MilevskyM. A. and PosnerS. E. (1998).Asian options, the sum of lognormals, and the reciprocal gamma distribution.J. Financial Quant. Anal. 33,409422.
[29] SchwartzS. C. and YehY.-S. (1982).On the distribution function and moments of power sums with log-normal components.Bell System Tech. J. 61,14411462.
[30] StehfestH. (1970).Algorithm 368: Numerical inversion of Laplace transforms [D5].Commun. ACM 13,4749.
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Advances in Applied Probability
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