Continuous-state branching processes with migration

• Autorzy: Vidmar Matija

Identyfikatory

DOI

10.37190/0208-4147.00044

• Języki publikacji: EN

Abstrakty

EN

Continuous-state branching processes (CSBPs) with immigration (CBIs), stopped on hitting zero, are generalized by allowing the process governing immigration to be any Lévy process without negative jumps. Unlike CBIs, these newly introduced processes do not appear to satisfy any natural affine property on the level of the Laplace transforms of the semigroups. Basic properties of these processes are described. Explicit formulae (on neighborhoods of infinity) for the Laplace transforms of the first passage times downwards and of the explosion time are derived.

Słowa kluczowe

EN

continuous-state branching process; stochastic differential equation; migration; first passage time; explosion; Laplace transform; scale function; Lamperti's time change; spectrally positive Lévy process

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2022
• Tom: Vol. 42, Fasc. 2
• Strony: 227--249
• Opis fizyczny: Bibliogr. 21 poz.

Twórcy

autor

Vidmar Matija

• Faculty of Mathematics and Physics, Department of Mathematics, University of Ljubljana, Jadranska ulica 21, 1000 Ljubljana, Slovenia

Bibliografia

• [1] M. E. Caballero, J. L. P. Garmendia and G. U. Bravo, A Lamperti-type representation of continuous-state branching processes with immigration, Ann. Probab. 41 (2013), 1585-1627.
• [2] M. E. Caballero, A. Lambert and G. U. Bravo, Proof(s) of the Lamperti representation of continuous-state branching processes, Probab. Surv. 6 (2009), 62-89.
• [3] X. Duhalde, C. Foucart and C. Ma, On the hitting times of continuous-state branching processes with immigration, Stochastic Process. Appl. 124 (2014), 4182-4201.
• [4] D. Fekete, J. Fontbona and A. E. Kyprianou, Skeletal stochastic differential equations for continuous-state branching processes, J. Appl. Probab. 56 (2019), 1122-1150.
• [5] W. Feller, An Introduction to Probability Theory and Its Applications, Volume 2, 2nd ed., Wiley, New York, 1971.
• [6] M. Fittipaldi and J. Fontbona, On SDE associated with continuous-state branching processes conditioned to never be extinct, Electron. Commun. Probab. 17 (2012), 1-13.
• [7] A. Göing-Jaeschke and M. Yor, A survey and some generalizations of Bessel processes, Bernoulli 9 (2003), 313-349.
• [8] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd ed., Kodansha, Tokyo, and North-Holland, Amsterdam, 1989.
• [9] J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, 2nd ed., Grundlehren Math. Wiss. 288, Springer, Berlin, 1987.
• [10] S. Jansen and N. Kurt, On the notion(s) of duality for Markov processes, Probab. Surv. 11 (2014), 59-120.
• [11] K. Kawazu and S. Watanabe, Branching processes with immigration and related limit theorems, Theory Probab. Appl. 16 (1971), 36-54.
• [12] A. E. Kyprianou, Fluctuations of Lévy Processes with Applications, 2nd ed., Universitext, Springer, Heidelberg, 2014.
• [13] J. Lamperti, Continuous state branching processes, Bull. Amer. Math. Soc. 73 (1967), 382-386.
• [14] H. Leman and J. C. Pardo, Extinction time of logistic branching processes in a Brownian environment, Latin Amer. J. Probab. Math. Statist. 18 (2021), 1859-1890.
• [15] Z. Li and F. Pu, Strong solutions of jump-type stochastic equations, Electron. Commun. Probab. 17 (2012), 1-13.
• [16] R. Ma, Lamperti transformation for continuous-state branching processes with competition and applications, Statist. Probab. Lett. 107 (2015), 11-17.
• [17] S. Palau and J. C. Pardo, Branching processes in a Lévy random environment, Acta Appl. Math. 153 (2018), 55-79.
• [18] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed., Grundlehren Math. Wiss. 293, Springer, Berlin, 1999.
• [19] R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions: Theory and Applications, 2nd ed., De Gruyter Stud. Math., De Gruyter, Berlin, 2012.
• [20] M. Vidmar, Some characterizations for Markov processes at first passage, arXiv:2112.11757v3(2021).
• [21] M. Vidmar, Some harmonic functions for killed Markov branching processes with immigration and culling, Stochastics 94 (2022), 578-601.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).