Distributional properties of jumps of multi-type CBI processes

Mátyás Barczy; Sandra Palau

doi:10.1214/24-EJP1125

2024 Distributional properties of jumps of multi-type CBI processes

Mátyás Barczy, Sandra Palau

Author Affiliations +

Electron. J. Probab. 29: 1-39 (2024). DOI: 10.1214/24-EJP1125

Abstract

We study the distributional properties of jumps of multi-type continuous state and continuous time branching processes with immigration (multi-type CBI processes). We derive an expression for the distribution function of the first jump time of a multi-type CBI process with jump size in a given Borel set having finite total Lévy measure, which is defined as the sum of the measures appearing in the branching and immigration mechanisms of the multi-type CBI process in question. Using this we derive an expression for the distribution function of the local supremum of the norm of the jumps of a multi-type CBI process. Further, we show that if A is a nondegenerate rectangle anchored at zero and with total Lévy measure zero, then the probability that the local coordinate-wise supremum of jumps of the multi-type CBI process belongs to A is zero. We also prove that a converse statement holds.

References

1.

Applebaum, D. (2009). Lévy Processes and Stochastic Calculus, 2nd ed. Cambridge University Press, Cambridge. MR2512800Applebaum, D. (2009). Lévy Processes and Stochastic Calculus, 2nd ed. Cambridge University Press, Cambridge. MR2512800

2.

Barczy, M., Li, Z. and Pap, G. (2015). Stochastic differential equation with jumps for multi-type continuous state and continuous time branching processes with immigration. ALEA. Latin American Journal of Probability and Mathematical Statistics 12(1) 129–169. MR3340375Barczy, M., Li, Z. and Pap, G. (2015). Stochastic differential equation with jumps for multi-type continuous state and continuous time branching processes with immigration. ALEA. Latin American Journal of Probability and Mathematical Statistics 12(1) 129–169. MR3340375

3.

Barczy, M. and Palau, S. (2023). On distributions of jumps of multi-type CBI processes. arXiv: 2308.05639.Barczy, M. and Palau, S. (2023). On distributions of jumps of multi-type CBI processes. arXiv: 2308.05639.

4.

Barczy, M., Palau, S. and Pap, G. (2020). Almost sure,

L_{1}

- and

L_{2}

-growth behavior of supercritical multi-type continuous state and continuous time branching processes with immigration. Science China Mathematics 63(10) 2089–2116. MR4159401Barczy, M., Palau, S. and Pap, G. (2020). Almost sure,

L_{1}

- and

L_{2}

-growth behavior of supercritical multi-type continuous state and continuous time branching processes with immigration. Science China Mathematics 63(10) 2089–2116. MR4159401

5.

Barczy, M., Palau, S. and Pap, G. (2021). Asymptotic behavior of projections of supercritical multi-type continuous-state and continuous-time branching processes with immigration. Advances in Applied Probability 53(4) 1023–1060. MR4342576Barczy, M., Palau, S. and Pap, G. (2021). Asymptotic behavior of projections of supercritical multi-type continuous-state and continuous-time branching processes with immigration. Advances in Applied Probability 53(4) 1023–1060. MR4342576

6.

Barczy, M. and Pap, G. (2016). Asymptotic behavior of critical, irreducible multi-type continuous state and continuous time branching processes with immigration. Stochastics and Dynamics 16(4) Article ID 1650008, 30 pages. MR3494683Barczy, M. and Pap, G. (2016). Asymptotic behavior of critical, irreducible multi-type continuous state and continuous time branching processes with immigration. Stochastics and Dynamics 16(4) Article ID 1650008, 30 pages. MR3494683

7.

Caballero, M. E., Pérez Garmendia, J. L., and Uribe Bravo, G. (2017). Affine processes on

R_{+}^{m} \times R^{n}

and multiparameter time changes. Annales de l’Institut Henri Poincaré – Probabilités et Statistiques 3(53) 1280–1304. MR3689968Caballero, M. E., Pérez Garmendia, J. L., and Uribe Bravo, G. (2017). Affine processes on

R_{+}^{m} \times R^{n}

and multiparameter time changes. Annales de l’Institut Henri Poincaré – Probabilités et Statistiques 3(53) 1280–1304. MR3689968

8.

Chaumont, L. and Marolleau, M. (2023). Extinction times of multitype continuous-state branching processes. Annales de l’Institut Henri Poincaré – Probabilités et Statistiques 59(2) 563–577. MR4575008Chaumont, L. and Marolleau, M. (2023). Extinction times of multitype continuous-state branching processes. Annales de l’Institut Henri Poincaré – Probabilités et Statistiques 59(2) 563–577. MR4575008

9.

Chen, S. and Li, Z. (2021). Continuous time mixed state branching processes and stochastic equations. Acta Mathematica Scientia 41B(5) 1445–1473. MR4279361Chen, S. and Li, Z. (2021). Continuous time mixed state branching processes and stochastic equations. Acta Mathematica Scientia 41B(5) 1445–1473. MR4279361

10.

Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Annals of Applied Probability 13 984–1053. MR1994043Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Annals of Applied Probability 13 984–1053. MR1994043

11.

Filipović, D., Mayerhofer, E. and Schneider, P. (2013). Density approximations for multivariate affine jump-diffusion processes. Journal of Econometrics 176(2) 93–111. MR3084047Filipović, D., Mayerhofer, E. and Schneider, P. (2013). Density approximations for multivariate affine jump-diffusion processes. Journal of Econometrics 176(2) 93–111. MR3084047

12.

Friesen, M., Jin, P. and Rüdiger, B. (2020). On the boundary behavior of multi-type continuous-state branching processes with immigration. Electronic Communications in Probability 25 1–14. MR4195178Friesen, M., Jin, P. and Rüdiger, B. (2020). On the boundary behavior of multi-type continuous-state branching processes with immigration. Electronic Communications in Probability 25 1–14. MR4195178

13.

Friesen, M., Jin, P. and Rüdiger, B. (2020). Existence of densities for multi-type continuous-state branching processes with immigration. Stochastic Processes and their Applications 130(9) 5426–5452. MR4127334Friesen, M., Jin, P. and Rüdiger, B. (2020). Existence of densities for multi-type continuous-state branching processes with immigration. Stochastic Processes and their Applications 130(9) 5426–5452. MR4127334

14.

He, X. and Li, Z. (2016). Distributions of jumps in a continuous-state branching process with immigration. Journal of Applied Probability 53 1166–1177. MR3581249He, X. and Li, Z. (2016). Distributions of jumps in a continuous-state branching process with immigration. Journal of Applied Probability 53 1166–1177. MR3581249

15.

Horn, R. A. and Johnson, Ch. R. (2013). Matrix Analysis, 2nd ed. Cambridge University Press, Cambridge. MR2978290Horn, R. A. and Johnson, Ch. R. (2013). Matrix Analysis, 2nd ed. Cambridge University Press, Cambridge. MR2978290

16.

Horst, U. and Xu, W. (2022). The microstructure of stochastic volatility models with self-exciting jump dynamics. The Annals of Applied Probability 32(6) 4568–4610. MR4522360Horst, U. and Xu, W. (2022). The microstructure of stochastic volatility models with self-exciting jump dynamics. The Annals of Applied Probability 32(6) 4568–4610. MR4522360

17.

Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Publishing Co., Amsterdam, Kodansha, Ltd., Tokyo. MR1011252Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Publishing Co., Amsterdam, Kodansha, Ltd., Tokyo. MR1011252

18.

Ji, L. and Li, Z. (2020). Moments of continuous-state branching processes with or without immigration. Acta Mathematicae Applicatae Sinica, English Series 36(2) 361–373. MR4083449Ji, L. and Li, Z. (2020). Moments of continuous-state branching processes with or without immigration. Acta Mathematicae Applicatae Sinica, English Series 36(2) 361–373. MR4083449

19.

Jiao, Y., Ma, C. and Scotti, S. (2017). Alpha-CIR model with branching processes in sovereign interest rate modeling. Finance and Stochastics 21 789–813. MR3663644Jiao, Y., Ma, C. and Scotti, S. (2017). Alpha-CIR model with branching processes in sovereign interest rate modeling. Finance and Stochastics 21 789–813. MR3663644

20.

Kallenberg, O. (2017). Random Measures, Theory and Applications. Springer, Cham. MR3642325Kallenberg, O. (2017). Random Measures, Theory and Applications. Springer, Cham. MR3642325

21.

Keller-Ressel, M. (2008). Affine Processes – Theory and Applications in Finance (Ph.D. thesis). Vienna University of Technology, 110 pages.Keller-Ressel, M. (2008). Affine Processes – Theory and Applications in Finance (Ph.D. thesis). Vienna University of Technology, 110 pages.

22.

Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press, New York. MR1207584Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press, New York. MR1207584

23.

Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications, 2nd ed. Springer-Verlag, Heidelberg. MR3155252Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications, 2nd ed. Springer-Verlag, Heidelberg. MR3155252

24.

Kyprianou, A. and Palau, S. (2018). Extinction properties of multi-type continuous-state branching processes. Stochastic Processes and their Applications 128(10) 3466–3489. MR3849816Kyprianou, A. and Palau, S. (2018). Extinction properties of multi-type continuous-state branching processes. Stochastic Processes and their Applications 128(10) 3466–3489. MR3849816

25.

Kyprianou, A., Palau, S. and Ren, Y. X. (2018). Almost sure growth of supercritical multi-type continuous-state branching process. ALEA 15 409–428. MR3795454Kyprianou, A., Palau, S. and Ren, Y. X. (2018). Almost sure growth of supercritical multi-type continuous-state branching process. ALEA 15 409–428. MR3795454

26.

Li, Z. (2022). Measure-Valued Branching Markov Processes, 2nd ed. Springer-Verlag GmbH, Berlin. MR4704078Li, Z. (2022). Measure-Valued Branching Markov Processes, 2nd ed. Springer-Verlag GmbH, Berlin. MR4704078

27.

Li, Z. (2020). Continuous-state branching processes with immigration. From probability to finance–lecture notes of BICMR Summer School on Financial Mathematics. Mathematical Lectures from Peking University, 1–69. Springer, Singapore. MR4339413Li, Z. (2020). Continuous-state branching processes with immigration. From probability to finance–lecture notes of BICMR Summer School on Financial Mathematics. Mathematical Lectures from Peking University, 1–69. Springer, Singapore. MR4339413

28.

Ma, R. G. (2013). Stochastic equations for two-type continuous-state branching processes with immigration. Acta Mathematica Sinica, English Series 29 287–294. MR3016529Ma, R. G. (2013). Stochastic equations for two-type continuous-state branching processes with immigration. Acta Mathematica Sinica, English Series 29 287–294. MR3016529

29.

Watanabe, S. (1969). On two dimensional Markov processes with branching property. Transactions of the American Mathematical Society 136 447–466. MR0234531Watanabe, S. (1969). On two dimensional Markov processes with branching property. Transactions of the American Mathematical Society 136 447–466. MR0234531

Creative Commons Attribution 4.0 International License.

Citation Download Citation

Mátyás Barczy and Sandra Palau "Distributional properties of jumps of multi-type CBI processes," Electronic Journal of Probability 29(none), 1-39, (2024). https://doi.org/10.1214/24-EJP1125

Received: 11 August 2023; Accepted: 15 April 2024; Published: 2024

Access the abstract

JOURNAL ARTICLE
39 PAGES

DOWNLOAD PDF + SAVE TO MY LIBRARY