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Strong feller and ergodic properties of the (1+1)-affine process

Part of: Markov processes Stochastic analysis

Published online by Cambridge University Press:  14 March 2023

Shukai Chen  and
Zenghu Li
Shukai Chen*
Affiliation:
Fujian Normal University
Zenghu Li*
Affiliation:
Beijing Normal University
*
*Postal address: School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350007, People’s Republic of China. Email address: [email protected]
**Postal address: Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China. Email address: [email protected]
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Abstract

We prove some estimates for the variations of transition probabilities of the (1+1)-affine process. From these estimates we deduce the strong Feller and the ergodic properties of the total variation distance of the process. The key strategy is the pathwise construction and analysis of several Markov couplings using strong solutions of stochastic equations.

Keywords

Affine processstrong Feller propertyergodicitytotal variation distancecoupling

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes 60H15: Stochastic partial differential equations
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 3 , September 2023 , pp. 812 - 834
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Bansaye, V. and Méléard, S. (2015). Stoch astic Models for Structured Populations: Scaling Limits and Long Time Behavior. Springer, Cham.Google Scholar
Bao, J. H. and Wang, J. (2020). Coupling methods and exponential ergodicity for two factor affine processes. Preprint. Available at https://arxiv.org/abs/2004.10384.Google Scholar
Barczy, M., Döring, L., Li, Z. and Pap, G. (2014). Stationarity and ergodicity for an affine two-factor model. Adv. Appl. Prob. 46, 878–898.10.1239/aap/1409319564CrossRefGoogle Scholar
Dawson, D. A. and Li, Z. (2006). Skew convolution semigroups and affine Markov processes. Ann. Prob. 34, 1103–1142.10.1214/009117905000000747CrossRefGoogle Scholar
Dawson, D. A. and Li, Z. (2012). Stochastic equations, flows and measure-valued processes. Ann. Prob. 40, 813–857.10.1214/10-AOP629CrossRefGoogle Scholar
Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Prob. 13, 984–1053.10.1214/aoap/1060202833CrossRefGoogle Scholar
Friesen, M. and Jin, P. (2020). On the anisotropic stable JCIR process. ALEA Lat. Amer. J. Prob. Math. Statist. 2, 643–674.10.30757/ALEA.v17-25CrossRefGoogle Scholar
Friesen, M., Jin, P. and Rüdiger, B. (2020). Stochastic equation and exponential ergodicity in Wasserstein distances for affine processes. Ann. Appl. Prob. 30, 2165–2195.10.1214/19-AAP1554CrossRefGoogle Scholar
Jin, P., Kremer, J. and Rüdiger, B. (2017). Exponential ergodicity of an affine two-factor model based on the α-root process. Adv. Appl. Prob. 49, 1144–1169.10.1017/apr.2017.37CrossRefGoogle Scholar
Jin, P., Kremer, J. and Rüdiger, B. (2019). Moments and ergodicity of the jump-diffusion CIR process. Stochastics 91, 974–997.10.1080/17442508.2019.1576686CrossRefGoogle Scholar
Jin, P., Kremer, J. and Rüdiger, B. (2020). Existence of limiting distribution for affine processes. J. Math. Anal. Appl. 486, article no. 123912, 31 pp.Google Scholar
Kawazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Theory Prob. Appl. 16, 36–54.10.1137/1116003CrossRefGoogle Scholar
Keller-Ressel, M., Schachermayer, W. and Teichmann, J. (2011). Affine processes are regular. Prob. Theory Relat. Fields 151, 591–611.10.1007/s00440-010-0309-4CrossRefGoogle Scholar
Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications, 2nd edn. Springer, Heidelberg.10.1007/978-3-642-37632-0CrossRefGoogle Scholar
Li, Z. (2011). Measure-Valued Branching Markov Processes. Springer, Heidelberg.10.1007/978-3-642-15004-3CrossRefGoogle Scholar
Li, Z. and Ma, C. (2015). Asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model. Stoch. Process. Appl. 125, 3196–3233.10.1016/j.spa.2015.03.002CrossRefGoogle Scholar
Li, Z. (2020). Continuous-state branching processes with immigration. In From Probability to Finance: Lecture Notes of BICMR Summer School on Financial Mathematics, ed. Y. Jiao, Springer, Singapore, pp. 1–69.Google Scholar
Li, Z. (2021). Ergodicities and exponential ergodicities of Dawson–Watanabe type processes. Theory Prob. Appl. 66, 276–298.10.1137/S0040585X97T990393CrossRefGoogle Scholar
Mayerhofer, E., Stelzer, R. and Vestweber, J. (2020). Geometric ergodicity of affine processes on cones. Stoch. Process. Appl. 130, 4141–4173.10.1016/j.spa.2019.11.012CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes. Adv. Appl. Prob. 25, 518–548.10.2307/1427522CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press.10.1017/CBO9780511626630CrossRefGoogle Scholar
Pardoux, E. (2016). Probabilistic Models of Population Evolution: Scaling Limits, Genealogies and Interactions. Springer, Cham.10.1007/978-3-319-30328-4CrossRefGoogle Scholar
Pinsky, M. A. (1972). Limit theorems for continuous state branching processes with immigration. Bull. Amer. Math. Soc. 78, 242–244.10.1090/S0002-9904-1972-12938-0CrossRefGoogle Scholar
Priola, E. and Zabczyk, J. (2009). Densities for Ornstein–Uhlenbeck processes with jumps. Bull. London Math. Soc. 41, 41–50.10.1112/blms/bdn099CrossRefGoogle Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.10.1007/978-3-662-06400-9CrossRefGoogle Scholar
Sato, K. and Yamazato, M. (1984). Operator-self-decomposable distributions as limit distributions of processes of Ornstein–Uhlenbeck type. Stoch. Process. Appl. 17, 73–100.10.1016/0304-4149(84)90312-0CrossRefGoogle Scholar
Schilling, R. L. and Wang, J. (2012). On the coupling property and the Liouville theorem for Ornstein–Uhlenbeck processes. J. Evol. Equat. 12, 119–140.10.1007/s00028-011-0126-yCrossRefGoogle Scholar
Wang, F. (2011). Coupling for Ornstein–Uhlenbeck jump processes with jumps. Bernoulli 17, 1136–1158.10.3150/10-BEJ308CrossRefGoogle Scholar
Wang, J. (2012). On the exponential ergodicity of Lévy-driven Ornstein–Uhlenbeck processes. J. Appl. Prob. 49, 990–1004.10.1239/jap/1354716653CrossRefGoogle Scholar
Zhang, X. and Glynn, P. (2018). Affine jump-diffusions: stochastic stability and limit theorems. Preprint. Available at https://arxiv.org/abs/1811.00122.Google Scholar