Exact Edgeworth expansion for a L\'{e}vy process
Heikki J. Tikanm\"aki
TL;DR
This paper derives an exact series representation for the distribution of certain Lévy processes and clarifies historical results on Edgeworth expansions for their distribution functions.
Contribution
It provides an exact series expansion for Lévy process distributions and clarifies Cramér's classical Edgeworth expansion results.
Findings
Exact series representation for Lévy process distributions
Clarification of Cramér's Edgeworth expansion results
Conditions under which the series applies
Abstract
The one dimensional distribution of a L\'{e}vy process is not known in general even though its characteristic function is given by the famous L\'{e}vy-Khinchine theorem. This article gives an exact series representation for the one dimensional distribution of a L\'{e}vy process satisfying certain moment conditions. Moreover, this work clarifies an old result by Cram\'{e}r on Edgeworth expansions for the distribution function of a L\'{e}vy process.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Stochastic processes and financial applications · Probability and Risk Models