Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Affine processes on $\mathbb{R}_{+}^{m}\times\mathbb{R}^{n}$ and multiparameter time changes

M. Emilia Caballero, José Luis Pérez Garmendia, and Gerónimo Uribe Bravo

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Abstract

We present a time change construction of affine processes with state-space $\mathbb{R}_{+}^{m}\times\mathbb{R}^{n}$. These processes were systematically studied in (Ann. Appl. Probab. 13 (2003) 984–1053) since they gather interesting classes of processes such as Lévy processes, continuous-state branching processes with immigration, and of the Ornstein–Uhlenbeck type. The construction is based on a (basically) continuous functional of a multidimensional Lévy process which implies that limit theorems for Lévy processes (both almost surely and in distribution) can be inherited to affine processes. The construction can be interpreted as a multiparameter time change scheme or as a (random) ordinary differential equation driven by discontinuous functions. In particular, we propose approximation schemes for affine processes based on the Euler method for solving the associated discontinuous ODEs, which are shown to converge.

Résumé

Nous présentons une construction des processus affines à valeurs dans $\mathbb{R}_{+}^{m}\times\mathbb{R}^{n}$ à partir de changement de temps. Ces processus ont été systématiquement étudiés dans (Ann. Appl. Probab. 13 (2003) 984–1053) car ils regroupent certaines classes intéressantes de processus tels que les processus de Lévy, les processus de branchement continu avec immigration et du type Ornstein–Uhlenbeck. La construction se base sur une fonctionnelle (presque) continue d’un processus de Lévy multidimensionnel, ce qui implique que les théorèmes limites pour les processus de Lévy (que ce soit presque sûrement ou en loi) peuvent être transmis aux processus affines. La construction peut être interprétée comme un changement de temps à plusieurs paramètres ou comme une équation différentielle ordinaire aléatoire dirigée par des fonctions discontinues. En particulier, on propose des schémas d’approximation pour les processus affines basés sur la méthode d’Euler pour résoudre les EDO discontinues associées, dont la convergence est démontrée.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1280-1304.

Dates
Received: 6 March 2015
Revised: 16 March 2016
Accepted: 20 March 2016
First available in Project Euclid: 21 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1500624042

Digital Object Identifier
doi:10.1214/16-AIHP755

Mathematical Reviews number (MathSciNet)
MR3689968

Zentralblatt MATH identifier
1378.60111

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F17: Functional limit theorems; invariance principles

Keywords
Lévy processes Continuous-state branching processes with immigration Ornstein–Uhlenbeck processes Multiparameter time change

Citation

Caballero, M. Emilia; Pérez Garmendia, José Luis; Uribe Bravo, Gerónimo. Affine processes on $\mathbb{R}_{+}^{m}\times\mathbb{R}^{n}$ and multiparameter time changes. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1280--1304. doi:10.1214/16-AIHP755. https://projecteuclid.org/euclid.aihp/1500624042


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References

  • [1] O. E. Barndorff-Nielsen and A. Shiryaev. Change of Time and Change of Measure. Advanced Series on Statistical Science & Applied Probability 13. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010.
  • [2] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge University Press, Cambridge, 1996.
  • [3] J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 (2) (2003) 261–288.
  • [4] M. E. Caballero, J. L. Pérez Garmendia and G. Uribe Bravo. A Lamperti-type representation of continuous-state branching processes with immigration. Ann. Probab. 41 (2013) 1585–1627.
  • [5] L. Chaumont. Breadth first search coding of multitype forests with application to Lamperti representation. In Séminaire de Probabilités XLVII—In Memoriam Marc Yor 561–584. C. Donati-Martin, A. Lejay and A. Rouault (Eds). Springer, Cham, 2015.
  • [6] L. Chaumont and R. Liu. Coding multitype forests: Application to the law of the total population of branching forests. Trans. Amer. Math. Soc. 368 (2016) 2723–2747.
  • [7] L. Chaumont and M. Yor. Exercises in Probability. Cambridge Series in Statistical and Probabilistic Mathematics 13. Cambridge University Press, Cambridge, 2003.
  • [8] C. Cuchiero and J. Teichmann. Path properties and regularity of affine processes on general state spaces. In Séminaire de Probabilités XLV 201–244. Lecture Notes in Math. 2078. Springer, Cham, 2013.
  • [9] D. A. Dawson and Z. Li. Skew convolution semigroups and affine Markov processes. Ann. Probab. 34 (3) (2006) 1103–1142.
  • [10] D. A. Dawson and Z. Li. Stochastic equations, flows and measure-valued processes. Ann. Probab. 40 (2) (2012) 813–857.
  • [11] D. Duffie, D. Filipović and W. Schachermayer. Affine processes and applications in finance. Ann. Appl. Probab. 13 (3) (2003) 984–1053.
  • [12] E. B. Dynkin. Markov Processes, Vols. I, II. Die Grundlehren der Mathematischen Wissenschaften 122. Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965.
  • [13] S. N. Ethier and T. G. Kurtz. Markov Processes. Characterization and Convergence. John Wiley & Sons Inc., New York, 1986.
  • [14] P. Friz and M. Hairer. A Course on Rough Paths. Universitext XIV. Springer, Cham, 2014.
  • [15] Z. Fu and Z. Li. Stochastic equations of non-negative processes with jumps. Stochastic Process. Appl. 120 (3) (2010) 306–330.
  • [16] N. Gabrielli. Affine processes from the perspective of path space valued Lévy processes. Ph.D. thesis, ETH, Zürich, 2014.
  • [17] N. Gabrielli and J. Teichmann. Pathwise construction of affine processes. Preprint, 2014. Available at arXiv:1412.7837.
  • [18] A. R. Galmarino. A test for Markov times. Rev. Un. Mat. Argentina 21 (1963) 173–178.
  • [19] A. Grimvall. On the convergence of sequences of branching processes. Ann. Probab. 2 (1974) 1027–1045.
  • [20] A. Joffe and M. Métivier. Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. in Appl. Probab. 18 (1) (1986) 20–65.
  • [21] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Springer-Verlag, New York, 2002.
  • [22] J. Kallsen. A didactic note on affine stochastic volatility models. In From Stochastic Calculus to Mathematical Finance 343–368. Springer, Berlin, 2006.
  • [23] K. Kiyoshi and S. Watanabe. Branching processes with immigration and related limit theorems. Teor. Veroyatn. Primen. 16 (1971) 34–51.
  • [24] M. Keller-Ressel, W. Schachermayer and J. Teichmann. Affine processes are regular. Probab. Theory Related Fields 151 (3–4) (2011) 591–611.
  • [25] M. Keller-Ressel, W. Schachermayer and J. Teichmann. Regularity of affine processes on general state spaces. Electron. J. Probab. 18 (2013) Art. ID 43.
  • [26] T. G. Kurtz. Representations of Markov processes as multiparameter time changes. Ann. Probab. 8 (4) (1980) 682–715.
  • [27] A. Lambert. Population dynamics and random genealogies. Stoch. Models 24 (Suppl. 1) (2008) 45–163.
  • [28] Z. Li. Branching processes with immigration and related topics. Front. Math. China 1 (1) (2006) 73–97.
  • [29] Z. Li. A limit theorem for discrete Galton–Watson branching processes with immigration. J. Appl. Probab. 43 (1) (2006) 289–295.
  • [30] P. E. Protter. Stochastic Integration and Differential Equations, 2nd edition. Springer-Verlag, Berlin, 2004.
  • [31] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften 293. Springer-Verlag, Berlin, 1999.
  • [32] B. A. Rogozin. The local behavior of processes with independent increments. Teor. Veroyatn. Primen. 13 (1968) 507–512.
  • [33] K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge University Press, Cambridge, 1999.
  • [34] M. Sharpe. General Theory of Markov Processes. Pure and Applied Mathematics 133. Academic Press, Inc., Boston, MA, 1988.
  • [35] T. Shiga and S. Watanabe. Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrsch. Verw. Gebiete 27 (1973) 37–46.
  • [36] V. A. Volkonskiĭ. Random substitution of time in strong Markov processes. Teor. Veroyatn. Primen. 3 (1958) 332–350.
  • [37] S. Watanabe. On two dimensional Markov processes with branching property. Trans. Amer. Math. Soc. 136 (1969) 447–466.
  • [38] W. Whitt. Some useful functions for functional limit theorems. Math. Oper. Res. 5 (1) (1980) 67–85.
  • [39] E. Wong and M. Zakai. On the relation between ordinary and stochastic differential equations. Internat. J. Engrg. Sci. 3 (1965) 213–229.
  • [40] B. Wu. On the weak convergence of subordinated systems. Statist. Probab. Lett. 78 (18) (2008) 3203–3211.