Approximating the Laplace transform of the sum of dependent lognormals

2015/07/14 by Patrick J. Laub, Laub, Patrick J., Søren Asmussen +5
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1507.03750

arxiv created 2015/09/07 · arxiv updated 2015/09/08

Abstract

Let (X₁, …, X_n) be multivariate normal, with mean vector \boldsymbolμ and covariance matrix \boldsymbolΣ, and S_n=e^X₁+⋯+e^X_n. The Laplace transform \cal L(θ)=𝔼e^-θS_n ∝ ∫ exp\-h_θ(\boldsymbolx)\ d \boldsymbolx is represented as \cal L(θ)I(θ), where \cal L(θ) is given in closed-form and I(θ) is the error factor (≈ 1). We obtain \cal L(θ) by replacing h_θ(\boldsymbolx) with a second order Taylor expansion around its minimiser \boldsymbolx^*. An algorithm for calculating the asymptotic expansion of \boldsymbolx^* is presented, and it is shown that I(θ)→ 1 as θ→∞. A variety of numerical methods for evaluating I(θ) are 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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