Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-22T08:21:48.888Z Has data issue: false hasContentIssue false
  • English
  • Français

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: 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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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