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On the Laplace transform of the Lognormal distribution

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Integral transforms of the lognormal distribution are of great importance in statistics and probability, yet closed-form expressions do not exist. A wide variety of methods have been employed to provide approximations, both analytical and numerical. In this paper, we analyze a closed-form approximation L˜(θ) of the Laplace transform L(θ) which is obtained via a modified version of Laplace's method. This approximation, given in terms of the Lambert W(⋅) function, is tractable enough for applications. We prove that L˜(θ) is asymptotically equivalent to L(θ) as θ→∞. We apply this result to construct a reliable Monte Carlo estimator of L(θ) and prove it to be logarithmically efficient in the rare event sense as θ→∞.
OriginalsprogEngelsk
UdgiverT.N. Thiele Centre, Department of Mathematics, Aarhus University
Antal sider25
StatusUdgivet - 2013
SerietitelThiele Research Reports
Nummer06

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2013-06

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