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arXiv · 0911.3449

Stochastic integral characterizations of semi-selfdecomposable distributions and related Ornstein-Uhlenbeck type processes

Makoto Maejima·Yohei Ueda
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Abstract

In this paper, three topics on semi-selfdecomposable distributions are studied. The first one is to characterize semi-selfdecomposable distributions by stochastic integrals with respect to Levy processes. This characterization defines a mapping from an infinitely divisible distribution with finite log-moment to a semi-selfdecomposable distribution. The second one is to introduce and study a Langevin type equation and the corresponding Ornstein-Uhlenbecktype process whose limiting distribution is semi-selfdecomposable. Also, semi-stationary Ornstein-Uhlenbeck type processes with semi-selfdecomposable distributions are constructed. The third one is to study the iteration of the mapping above. The iterated mapping is expressed as a single mapping with a different integrand. Also, nested subclasses of the class of semi-selfdecomposable distributions are considered, andit is shown that the limit of these nested subclasses is the closure of the class of semi-stable distributions.

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Makoto Maejima, Yohei Ueda. 2009-11-18. Stochastic integral characterizations of semi-selfdecomposable distributions and related Ornstein-Uhlenbeck type processes. https://arxiv.org/abs/0911.3449

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