Inversions of Infinitely Divisible Distributions and Conjugates of Stochastic Integral Mappings

Ken-iti Sato ()
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Ken-iti Sato: Hachiman-yama 1101-5-103

Journal of Theoretical Probability, 2013, vol. 26, issue 4, 901-931

Abstract: Abstract The dual of an infinitely divisible distribution on ℝ d without Gaussian part defined in Sato (ALEA Lat. Am. J. Probab. Math. Statist. 3:67–110, 2007) is renamed to the inversion. Properties and characterization of the inversion are given. A stochastic integral mapping is a mapping μ=Φ f ρ of ρ to μ in the class of infinitely divisible distributions on ℝ d , where μ is the distribution of an improper stochastic integral of a nonrandom function f with respect to a Lévy process on ℝ d with distribution ρ at time 1. The concept of the conjugate is introduced for a class of stochastic integral mappings and its close connection with the inversion is shown. The domains and ranges of the conjugates of three two-parameter families of stochastic integral mappings are described. Applications to the study of the limits of the ranges of iterations of stochastic integral mappings are made.

Keywords: Infinitely divisible distribution; Inversion; Stochastic integral mapping; Conjugate; Monotone of order p; Increasing of order p; Class L ∞; 60E07; 60G51; 60H05 (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10959-012-0420-9

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