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Generalised Ornstein-Uhlenbeck Processes and the Convergence of Lévy Integrals

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Séminaire de Probabilités XXXVIII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 1857))

Abstract

Exponential functionals of the form \( \int_{0}^t \mathrm{e}^{-\xi_{s-}} \mathrm{d}\eta_s\) constructed from a two dimensional Lévy process \((\xi,\eta)\) are of interest and application in many areas. In particular, the question of the convergence of the integral \( \int_{0}^\infty \mathrm{e}^{-\xi_{t-}} \mathrm{d}\eta_t\) arises in recent investigations such as those of Barndorff-Nielsen and Shephard [3] in financial econometrics, and in those of Carmona, Petit and Yor [9], and Yor [40, 41], where it is related among other things to the existence of an invariant measure for a generalised Ornstein-Uhlenbeck process. We give a complete solution to the convergence question for integrals of the form \(\int_0^\infty g(\xi_{t-}) \mathrm{d}\eta_t\), when g(t) = e-t and \(\eta_t\) is general, or \(g(\cdot)\) is a nonincreasing function and \(\mathrm{d}\eta_t = \mathrm{d} t\), and some other related results. The necessary and sufficient conditions for convergence are stated in terms of the canonical characteristics of the Lévy process. Some applications in various areas (compound Poisson processes, subordinated perpetuities, the Doléans-Dade exponential) are also outlined.

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Correspondence to K. Bruce Erickson .

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Michel Émery Michel Ledoux Marc Yor

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© 2005 Springer-Verlag Berlin/Heidelberg

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Erickson, K.B., Maller, R.A. (2005). Generalised Ornstein-Uhlenbeck Processes and the Convergence of Lévy Integrals. In: Émery, M., Ledoux, M., Yor, M. (eds) Séminaire de Probabilités XXXVIII. Lecture Notes in Mathematics, vol 1857. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31449-3_6

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