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Theory of Probability and Mathematical Statistics

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For which functions are $f(X_t)-\mathbb {E} f(X_t)$ and $g(X_t)/\mathbb {E} g(X_t)$ martingales?


Authors: F. Kühn and R. L. Schilling
Journal: Theor. Probability and Math. Statist. 105 (2021), 79-91
MSC (2020): Primary 60G44, 60G51, 60J65; Secondary 39B22, 45E10
DOI: https://doi.org/10.1090/tpms/1157
Published electronically: December 7, 2021
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $X=(X_t)_{t\geq 0}$ be a one-dimensional Lévy process such that each $X_t$ has a $C^1_b$-density w. r. t. Lebesgue measure and certain polynomial or exponential moments. We characterize all polynomially bounded functions $f\colon \mathbb {R}\to \mathbb {R}$, and exponentially bounded functions $g\colon \mathbb {R}\to (0,\infty )$, such that $f(X_t)-\mathbb {E} f(X_t)$, resp. $g(X_t)/\mathbb {E} g(X_t)$, are martingales.


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Additional Information

F. Kühn
Affiliation: Fakultät Mathematik, Institut für Mathematische Stochastik, TU Dresden, 01062 Dresden, Germany
Email: franziska.kuehn1@tu-dresden.de

R. L. Schilling
Affiliation: Fakultät Mathematik, Institut für Mathematische Stochastik, TU Dresden, 01062 Dresden, Germany
Email: rene.schilling@tu-dresden.de

Keywords: Lévy process, Brownian motion, martingale, polynomial process, convolution equation, Choquet–Deny theorem, Cauchy functional equation, harmonic polynomial
Received by editor(s): August 5, 2021
Published electronically: December 7, 2021
Additional Notes: Financial support through the joint Polish–German NCN–DFG ‘Beethoven Classic 3’ grant (NCN 2018/31/G/ST1/02252; DFG SCHI 419/11-1) is gratefully acknowledged
Article copyright: © Copyright 2021 Taras Shevchenko National University of Kyiv