## Electronic Communications in Probability

### On the boundary behavior of multi-type continuous-state branching processes with immigration

#### Abstract

In this article, we provide a sufficient condition for a continuous-state branching process with immigration (CBI process) to not hit its boundary, i.e. for non-extinction. Our result applies to arbitrary dimension $d\geq 1$ and is formulated in terms of an integrability condition for its immigration and branching mechanisms $F$ and $R$. The proof is based on a comparison principle for multi-type CBI processes being compared to one-dimensional CBI processes, and then an application of an existing result for one-dimensional CBI processes. The same technique is also used to provide a sufficient condition for the transience of multi-type CBI processes.

#### Article information

Source
Electron. Commun. Probab., Volume 25 (2020), paper no. 84, 14 pp.

Dates
Accepted: 18 November 2020
First available in Project Euclid: 24 December 2020

https://projecteuclid.org/euclid.ecp/1608779421

Digital Object Identifier
doi:10.1214/20-ECP364

#### Citation

Friesen, Martin; Jin, Peng; Rüdiger, Barbara. On the boundary behavior of multi-type continuous-state branching processes with immigration. Electron. Commun. Probab. 25 (2020), paper no. 84, 14 pp. doi:10.1214/20-ECP364. https://projecteuclid.org/euclid.ecp/1608779421

#### References

• [1] Aurélien Alfonsi, Affine diffusions and related processes: simulation, theory and applications, Bocconi & Springer Series, vol. 6, Springer, Cham; Bocconi University Press, Milan, 2015.
• [2] Mátyás Barczy, Zenghu Li, and Gyula Pap, Stochastic differential equation with jumps for multi-type continuous state and continuous time branching processes with immigration, ALEA Lat. Am. J. Probab. Math. Stat. 12 (2015), no. 1, 129–169.
• [3] M. Emilia Caballero, José Luis Pérez Garmendia, and Gerónimo Uribe Bravo, A Lamperti-type representation of continuous-state branching processes with immigration, Ann. Probab. 41 (2013), no. 3A, 1585–1627.
• [4] M. Emilia Caballero, José Luis Pérez Garmendia, and Gerónimo Uribe Bravo, A Lamperti-type representation of continuous-state branching processes with immigration, Ann. Probab. 41 (2013), no. 3A, 1585–1627.
• [5] M. Emilia Caballero, José Luis Pérez Garmendia, and Gerónimo Uribe Bravo, Affine processes on $\mathbb {R}_{+}^{m}\times \mathbb {R}^{n}$ and multiparameter time changes, Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017), no. 3, 1280–1304.
• [6] Marie Chazal, Ronnie Loeffen, and Pierre Patie, Smoothness of continuous state branching with immigration semigroups, J. Math. Anal. Appl. 459 (2018), no. 2, 619–660.
• [7] Darrell Duffie, Damir Filipovic, and Walter Schachermayer, Affine processes and applications in finance, Ann. Appl. Probab. 13 (2003), no. 3, 984–1053.
• [8] Xan Duhalde, Clément Foucart, and Chunhua Ma, On the hitting times of continuous-state branching processes with immigration, Stochastic Process. Appl. 124 (2014), no. 12, 4182–4201.
• [9] Stewart Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiles & Sons, Inc., New York, 1986, Characterization and convergence.
• [10] William Feller, Diffusion processes in genetics, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, 1951, pp. 227–246.
• [11] Damir Filipovic, Eberhard Mayerhofer, and Paul Schneider, Density approximations for multivariate affine jump-diffusion processes, J. Econometrics 176 (2013), no. 2, 93–111.
• [12] Clément Foucart and Gerónimo Uribe Bravo, Local extinction in continuous-state branching processes with immigration, Bernoulli 20 (2014), no. 4, 1819–1844.
• [13] Martin Friesen and Peng Jin, On the anisotropic stable JCIR process, ALEA Lat. Am. J. Probab. Math. Stat. 17 (2020), no. 2, 643–674.
• [14] Martin Friesen, Peng Jin, and Barbara Rüdiger, Existence of densities for multi-type continuous-state branching processes with immigration, Stochastic Process. Appl. 130 (2020), no. 9, 5426–5452.
• [15] Martin Friesen, Peng Jin, and Barbara Rüdiger, Existence of densities for stochastic differential equations driven by a Lévy process with anisotropic jumps, (to appear in AIHP) (2020).
• [16] Martin Friesen, Peng Jin, and Barbara Rüdiger, Stochastic equation and exponential ergodicity in Wasserstein distances for affine processes, Ann. Appl. Probab. 30 (2020), no. 5, 2165–2195.
• [17] Zongfei Fu and Zenghu Li, Stochastic equations of non-negative processes with jumps, Stochastic Process. Appl. 120 (2010), no. 3, 306–330.
• [18] Peter W. Glynn and Xiaowei Zhang, Affine jump-diffusions: Stochastic stability and limit theorems, arXiv:1811.00122 (2018).
• [19] D. R. Grey, Asymptotic behaviour of continuous time, continuous state-space branching processes, J. Appl. Probability 11 (1974), 669–677.
• [20] Anders Grimvall, On the convergence of sequences of branching processes, Ann. Probability 2 (1974), 1027–1045.
• [21] Nobuyuki Ikeda and Shinzo Watanabe, Stochastic Differential Equations and Diffusion Processes, 2. ed. ed., North Holland mathematical library, Teil 24, North-Holland Publ. Co. u.a., Amsterdam u.a., 1989.
• [22] Peng Jin, Jonas Kremer, and Barbara Rüdiger, Existence of limiting distribution for affine processes, J. Math. Anal. Appl. 486 (2020), no. 2, 123912, 31.
• [23] Miloslav Jirina, Stochastic branching processes with continuous state space, Czechoslovak Math. J. 8 (83) (1958), 292–313.
• [24] A. Joffe and M. Métivier, Weak convergence of sequences of semimartingales with applications to multitype branching processes, Adv. in Appl. Probab. 18 (1986), no. 1, 20–65.
• [25] Kamil Kaleta and Pawel Sztonyk, Small-time sharp bounds for kernels of convolution semigroups, J. Anal. Math. 132 (2017), 355–394.
• [26] Kiyoshi Kawazu and Shinzo Watanabe, Branching processes with immigration and related limit theorems, Teor. Verojatnost. i Primenen. 16 (1971), 34–51.
• [27] Andreas E. Kyprianou, Introductory lectures on fluctuations of Lévy processes with applications, Universitext, Springer-Verlag, Berlin, 2006.
• [28] John Lamperti, The limit of a sequence of branching processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 7 (1967), 271–288.
• [29] Zenghu Li, A limit theorem for discrete Galton-Watson branching processes with immigration, J. Appl. Probab. 43 (2006), no. 1, 289–295.
• [30] Zenghu Li, Measure-valued branching Markov processes, Probability and its Applications (New York), Springer, Heidelberg, 2011.
• [31] Ru Gang Ma, Stochastic equations for two-type continuous-state branching processes with immigration, Acta Math. Sin. (Engl. Ser.) 29 (2013), no. 2, 287–294.
• [32] Eberhard Mayerhofer, Robert Stelzer, and Johanna Vestweber, Geometric ergodicity of affine processes on cones, Stochastic Process. Appl. 130 (2020), no. 7, 4141–4173.
• [33] Étienne Pardoux, Probabilistic models of population evolution, Mathematical Biosciences Institute Lecture Series. Stochastics in Biological Systems, vol. 1, Springer, [Cham]; MBI Mathematical Biosciences Institute, Ohio State University, Columbus, OH, 2016, Scaling limits, genealogies and interactions.
• [34] Shinzo Watanabe, On two dimensional Markov processes with branching property, Trans. Amer. Math. Soc. 136 (1969), 447–466.
• [35] Toshiro Watanabe, Sato’s conjecture on recurrence conditions for multidimensional processes of Ornstein-Uhlenbeck type, J. Math. Soc. Japan 50 (1998), no. 1, 155–168.