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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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.
References
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References
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- K.-S. Lau and C. R. Rao, Integrated Cauchy functional equation and characterizations of the exponential law, Sankhya, Ser. A 44 (1982), 72–90. MR 753078
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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
