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Approximating the Laplace transform of the sum of dependent lognormals

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

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
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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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