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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

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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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
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
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