Lévy processes, generalized moments and uniform integrability

• Autorzy: Berger David , Kühn Franziska , Schilling René L

Identyfikatory

DOI

10.37190/0208-4147.00045

• Języki publikacji: EN

Abstrakty

EN

We give new proofs of certain equivalent conditions for the existence of generalized moments of a Lévy process (X_t)_t≥0; in particular, the existence of a generalized g-moment is equivalent to the uniform integrability of (g(X_t))_{t Є [0,1]}. As a consequence, certain functions of a Lévy process which are integrable and local martingales are already true martingales. Our methods extend to moments of stochastically continuous additive processes, and we give new, short proofs for the characterization of lattice distributions and the transience of Leâvy processes.

Słowa kluczowe

EN

Lévy process; additive process; Dynkin's formula; generalized moment; Gronwall's inequality; local martingale; condition D; condition DL

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2022
• Tom: Vol. 42, Fasc. 1
• Strony: 109--131
• Opis fizyczny: Bibliogr. 15 poz.

Twórcy

autor

Berger David

• Institut für Mathematische Stochastik, Fakultät Mathematik, TU Dresden 01062, Dresden, Germany

autor

Kühn Franziska

• Institut für Mathematische Stochastik, Fakultät Mathematik, TU Dresden 01062, Dresden, Germany

autor

Schilling René L

• Institut für Mathematische Stochastik, Fakultät Mathematik, TU Dresden 01062, Dresden, Germany

Bibliografia

• 1. B. Böttcher, R. L. Schilling and J. Wang, Lévy-Type Processes: Construction, Approximation and Sample Path Properties, Lecture Notes in Math. 2099, Springer, Berlin, 2014.
• 2. C. Deng and R. L. Schilling, On shift Harnack inequalities for subordinate semigroups and moment estimates for éLvy processes, Stoch. Process. Appl. 125 (2015), 3851-3878.
• 3. T. Fujiwara, On the exponential moments of additive processes, J. Stoch. Anal. 2 (2021), art. 11, 21 pp.
• A. Hulanicki, A class of convolution semi-groups of measures on a Lie group, in: A. Weron (ed.), Probability Theory on Vector Spaces, Lecture Notes in Math. 828, Springer, Berlin 1980, 82-101.
• 4. N. Jacob, Pseudo Differential Operators and Markov Processes. Vol. 1, Imperial College Press, London, 2001.
• 5. D. Khoshnevisan and R. L. Schilling, From Lévy-Type Processes to Parabolic SPDEs, Adv. Courses Math. CRM Barcelona, Birkhäuser, Cham, 2017.
• 6. M. J. Klass and M. Yang, Maximal inequalities for additive processes, J. Theor. Probab. 25 (2012), 981-1012.
• 7. F. Kühn, Existence and estimates of moments for Lévy-type processes, Stoch. Process. Appl. 127 (2017), 1018-1041.
• 8. F. Kühn and R. L. Schilling, Strong convergence of the Euler–Maruyama approximation for a class of Lévy-driven SDEs, Stoch. Process. Appl. 129 (2019), 2654-2680.
• 9. E. Lukacs, Characteristic Functions, 2nd ed., Griffin, London, 1970.
• 10. D. Revuz na M. Yor, Continuous Martingales and Brownian Motion, 3rd ed., Springer, Berlin, 1999.
• 11. K. Sato, Lévy Processes and Infinitely Divisible Distributions, rev. ed., Cambridge Univ. Press, Cambridge, 2013.
• 12. R. L. Schilling, Measures, Integrals and Martingales, 2nd ed., Cambridge Univ. Press, Cambridge, 2017.
• 13. R. L. Schilling, Brownian Motion, 3rd ed., De Gruyter, Berlin, 2021.
• 14. E. Siebert, Continuous convolution semigroups integrating a submultiplicative function, Manuscripta Math. 37 (1982), 383-391.
• 15. H. Zhang, M. Zhao and J. Ying, α -transience and α-recurrence for random walks and Lévy processes, Chinese Ann. Math. 26B (2005), 127-142.