Theory of Probability and Mathematical Statistics
For which functions are $f(X_t)-\Ee f(X_t)$ and $g(X_t)/\Ee g(X_t)$ martingales?
F. K\"{u}hn, R. L. Schilling
Link
Abstract: Let $X=(X_t)_{t\geq 0}$ be a one-dimensional L\'{e}vy 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 $f\colon\real\to\real$, and exponentially bounded functions $g\colon\real\to (0,\infty)$, such that $f(X_t)-\Ee f(X_t)$, resp.\ $g(X_t)/\Ee g(X_t)$, are martingales.
Keywords: L\'{e}vy process, Brownian motion, martingale, polynomial process, convolution equation, Choquet--Deny theorem, Cauchy functional equation, harmonic polynomial
Bibliography: 1. J. Acz\'el, Lectures on Functional Equations and Their Applications, Mathematics in Science and Engineering vol. 19, Academic Press, New York, 1966.
2. Th. Anghelutza, Sur une \'equation fonctionnelle caract\'erisant les polynomes, Matematica (Cluj) 6 (1932), 1–7.
3. D. Berger, F. K\"{u}hn, and R. L. Schilling, L\'{e}vy processes, martingales and uniform integrability, preprint arXiv:2102.09004 [math.PR].
4. D. Berger and R. L. Schilling, On the Liouville and strong Liouville properties for a class of non-local operators, Math. Scand (to appear), preprint arXiv:2101.01592 [math.PR].
5. P. L. Butzer and W. Kozakiewicz, On the Riemann derivatives for integrable functions, Bull. Amer. Math. Soc. 59 (1953), 572–581.
6. G. Choquet and J. Deny, Sur l'\'equation de convolution $\mu=\mu \ast \sigma$, C. R. Acad. Sci. Paris 250 (1960), 799–801.
7. J. Deny, Sur l'\'equation de convolution $\mu=\mu*\sigma$}, S\'em.\ Brelot--Choquet--Deny. Th\'eorie du potentiel 4 (1959–1960), 1–11.
8. V. Knopova and R. L. Schilling, A note on the existence of transition probability densities for L\'{e}vy processes, Forum Math. 25 (2013), 125–149.
9. F. K\"{u}hn, A Liouville theorem for L\'{e}vy generators, Positivity 25 (2021), 997–1012.
10. K.-S. Lau and C. R. Rao, Integrated Cauchy functional equation and characterizations of the exponential law, Sankhya, Ser. A 44 (1982), 72–90.
11. M. Mania and R. Tevzadze, On martingale transformation of the linear Brownian motion, Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 34 (2020), 58–61.
12. M. Mania and R. Tevzadze, On martingale transformations of multidimensional Brownian motion, Statist. Probab. Lett. 175 (2021), Article No. 109119.
13. M. Mania and R. Tevzadze, Functional equations and martingales, Aequat. Math (to appear), preprint arXiv:1912.06299 [math.PR].
14. B. Ramachandran and B. L. S. Prakasa Rao, On the equation $f(x) = \int_{-\infty}^{\infty} f(x+y) \, d\mu(y)$, Sankhya, Ser. A 46 (1984), 326–338.
15. D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, third edition, Grundlehren Math. Wiss. vol. 293, Springer, Berlin, 1999.
16. W. Rudin, Functional Analysis. Second edition, International Series in Pure and Applied Mathematics, McGraw–Hill, New York, 1991.
17. K. Sato, L\'{e}vy Processes and Infinitely Divisible Distributions. Second edition, Cambridge Studies in Advanced Mathematics vol. 68, Cambridge University Press, Cambridge, 2013.
18. R. L. Schilling, An introduction to L\'{e}vy and Feller processes, in: D. Khoshnevisan and R. L. Schilling, From L\'{e}vy-type Processes to Parabolic SPDEs, Adv. Courses Math. CRM Barcelona, Birkh\"{a}use/Springer, Cham, 2016, pp. 1–97.
19. M. Sharpe, Zeroes of infinitely divisible densities, Ann. Math. Statist. 40 (1969), 1503–1505.
20. A. N. Shiryaev, Probability, third edition, Graduate Texts in Mathematics vol. 95, Springer, New York, 2016.
