Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-28T00:22:52.006Z Has data issue: false hasContentIssue false
  • English
  • Français

Stationarity and Ergodicity for an Affine Two-Factor Model

Part of: Markov processes Ergodic theory

Published online by Cambridge University Press:  22 February 2016

Mátyás Barczy ,
Leif Döring ,
Zenghu Li  and
Gyula Pap
Mátyás Barczy*
Affiliation:
University of Debrecen
Leif Döring*
Affiliation:
Universität Zürich
Zenghu Li*
Affiliation:
Beijing Normal University
Gyula Pap*
Affiliation:
University of Szeged
*
∗ Postal address: Faculty of Informatics, University of Debrecen, Pf. 12, H-4010 Debrecen, Hungary. Email address: barczy.matyas@inf.unideb.hu
∗∗ Postal address: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland.
∗∗∗ Postal address: School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China.
∗∗∗∗ Postal address: Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the existence of a unique stationary distribution and ergodicity for a two-dimensional affine process. Its first coordinate process is supposed to be a so-called α-root process with α ∈ (1, 2]. We prove the existence of a unique stationary distribution for the affine process in the α ∈ (1, 2] case; furthermore, we show ergodicity in the α = 2 case.

Keywords

Affine processstationary distributionergodicityFoster-Lyapunov criteria

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 37A25: Ergodicity, mixing, rates of mixing
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 3 , September 2014 , pp. 878 - 898
Copyright
© Applied Probability Trust 

References

Andersen, L. B. G. and Piterbarg, V. V. (2007). Moment explosions in stochastic volatility models. Finance Stoch. 11, 29–50.Google Scholar
Barczy, M., Döring, L., Li, Z. and Pap, G. (2013). On parameter estimation for critical affine processes. Electron. J. Statist. 7, 647–696.CrossRefGoogle Scholar
Barczy, M., Döring, L., Li, Z. and Pap, G. (2013). Stationarity and ergodicity for an affine two factor model. Preprint. Available at http://uk.arxiv.org/abs/1302.2534v4.Google Scholar
Barndorff-Nielsen, O. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. R. Statist. Soc. B 63, 167–241.Google Scholar
Ben Alaya, M. and Kebaier, A. (2013). Asymptotic behavior of the maximum likelihood estimator for ergodic and nonergodic square-root diffusions. Stoch. Anal. Appl. 31, 552–573.Google Scholar
Bhattacharya, R. N. (1982). On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Z. Wahrscheinlichkeitsth. 60, 185–201.Google Scholar
Carr, P. and Wu, L. (2003). The finite moment log stable process and option pricing. J. Finance 58, 753–777.Google Scholar
Chen, H. and Joslin, S. (2012). Generalized transform analysis of affine processes and applications in finance. Rev. Financial Stud. 25, 2225–2256.CrossRefGoogle Scholar
Chung, K. L. (1982). Lectures from Markov Processes to Brownian Motion. Springer, New York.Google Scholar
Cox, J. C., Ingersoll, J. E. Jr. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53, 385–407.Google Scholar
Cuchiero, C., Keller-Ressel, M. and Teichmann, J. (2012). Polynomial processes and their applications to mathematical finance. Finance Stoch. 16, 711–740.Google Scholar
Dawson, D. A. and Li, Z. (2006). Skew convolution semigroups and affine Markov processes. Ann. Prob. 34, 1103–1142.Google Scholar
Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Prob. 13, 984–1053.Google Scholar
Filipović, D. and Mayerhofer, E. (2009). Affine diffusion processes: theory and applications. In Advanced Financial Modelling (Radon Ser. Comput. Appl. Math. 8), De Gruyter, Berlin, pp. 125–164.Google Scholar
Filipović, D., Mayerhofer, E. and Schneider, P. (2013). Density approximations for multivariate affine Jump-diffusion processes. J. Econometrics 176, 93–111.CrossRefGoogle Scholar
Fu, Z. and Li, Z. (2010). Stochastic equations of non-negative processes with Jumps. Stoch. Process. Appl. 120, 306–330.Google Scholar
Glasserman, P. and Kim, K.-K. (2010). Moment explosions and stationary distributions in affine diffusion models. Math. Finance 20, 1–33.Google Scholar
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Stud. 6, 327–343.Google Scholar
Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.Google Scholar
Jeanblanc, M., Yor, M. and Chesney, M. (2009). Mathematical Methods for Financial Markets. Springer, London.CrossRefGoogle Scholar
Jena, R. P., Kim, K.-K. and Xing, H. (2012). Long-term and blow-up behaviors of exponential moments in multi-dimensional affine diffusions. Stoch. Process. Appl. 122, 2961–2993.Google Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York.Google Scholar
Keller-Ressel, M. and Mijatović, A. (2012). On the limit distributions of continuous-state branching processes with immigration. Stoch. Process. Appl. 122, 2329–2345.Google Scholar
Keller-Ressel, M., Schachermayer, W. and Teichmann, J. (2011). Affine processes are regular. Prob. Theory Relat. Fields 151, 591–611.Google Scholar
Li, Z. (2011). Measure-Valued Branching Markov Processes. Springer, Heidelberg.Google Scholar
Li, Z. and Ma, C. (2013). Asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model. Preprint. Available at http://uk.arxiv.org/abs/1301.3243.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1992). Stability of Markovian processes. I. Criteria for discrete-time chains. Adv. Appl. Prob. 24, 542–574.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. II. Continuous-time processes and sampled chains. Adv. Appl. Prob. 25, 487–517.Google 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.Google Scholar
Meyn, S. P. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Sandrić, N. (2013). Long-time behavior of stable-like processes. Stoch. Process. Appl. 123, 1276–1300.Google Scholar
Sharpe, M. (1988). General Theory of Markov Processes. Academic Press, Boston, MA.Google Scholar
Stroock, D. W. (1993). Probability Theory, An Analytic View. Cambridge University Press.Google Scholar
Volkmann, P. (1972). Gewöhnliche Differentialungleichungen mit quasimonoton wachsenden funktionen in topologischen Vektorräumen. Math. Z. 127, 157–164.Google Scholar
Zolotarev, V. M. (1986). One-Dimensional Stable Distributions (Transl. Math. Monogr. 65). American Mathematical Society, Providence, RI.Google Scholar