Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T18:59:22.278Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential ergodicity of an affine two-factor model based on the α-root process

Part of: Markov processes Ergodic theory

Published online by Cambridge University Press:  17 November 2017

Peng Jin ,
Jonas Kremer  and
Barbara Rüdiger
Peng Jin*
Affiliation:
Bergische Universität Wuppertal
Jonas Kremer*
Affiliation:
Bergische Universität Wuppertal
Barbara Rüdiger*
Affiliation:
Bergische Universität Wuppertal
*
* Postal address: Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, 42119 Wuppertal, Germany.
* Postal address: Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, 42119 Wuppertal, Germany.
* Postal address: Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, 42119 Wuppertal, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study an affine two-factor model introduced by Barczy et al. (2014). One component of this two-dimensional model is the so-called α-root process, which generalizes the well-known Cox–Ingersoll–Ross process. In the α = 2 case, this two-factor model was used by Chen and Joslin (2012) to price defaultable bonds with stochastic recovery rates. In this paper we prove exponential ergodicity of this two-factor model when α ∈ (1, 2). As a possible application, our result can be used to study the parameter estimation problem of the model.

Keywords

Affine processexponential ergodicityα-root processtransition densityFoster–Lyapunov function

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces 37A25: Ergodicity, mixing, rates of mixing
Secondary: 60J35: Transition functions, generators and resolvents 60J75: Jump processes
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 4 , December 2017 , pp. 1144 - 1169
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alaya, M. B. and Kebaier, A. (2012). Parameter estimation for the square-root diffusions: ergodic and nonergodic cases. Stoch. Models 28, 609–634. Google Scholar
[2] Barczy, M. and Pap, G. (2016). Asymptotic properties of maximum-likelihood estimators for Heston models based on continuous time observations. Statistics 50, 389–417. Google Scholar
[3] Barczy, M., Döring, L., Li, Z. and Pap, G. (2014). Parameter estimation for a subcritical affine two factor model. J. Statist. Planning Inference 151/152, 37–59. CrossRefGoogle Scholar
[4] 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. Google Scholar
[5] Chen, H. and Joslin, S. (2012). Generalized transform analysis of affine processes and applications in finance. Rev. Financial Studies 25, 2225–2256. Google Scholar
[6] Cox, J. C., Ingersoll, J. E. Jr. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53, 385–408. CrossRefGoogle Scholar
[7] Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Prob. 13, 984–1053. Google Scholar
[8] Duffie, D., Pan, J. and Singleton, K. (2000). Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376. Google Scholar
[9] Duhalde, X., Foucart, C. and Ma, C. (2014). On the hitting times of continuous-state branching processes with immigration. Stoch. Process. Appl. 124, 4182–4201. CrossRefGoogle Scholar
[10] Fournier, N. (1999). Strict positivity of the density for a Poisson driven S.D.E. Stoch. Stoch. Reports 68, 1–43. CrossRefGoogle Scholar
[11] Freitag, E. and Busam, R. (2009). Complex Analysis, 2nd edn. Springer, Berlin. Google Scholar
[12] Fu, Z. and Li, Z. (2010). Stochastic equations of non-negative processes with jumps. Stoch. Process. Appl. 120, 306–330. CrossRefGoogle Scholar
[13] Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327–343. Google Scholar
[14] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes (North-Holland Math. Library 24), 2nd edn. North-Holland, Amsterdam. Google Scholar
[15] Jin, P., Rüdiger, B. and Trabelsi, C. (2016). Exponential ergodicity of the jump-diffusion CIR process. In Stochastics of Environmental and Financial Economics, Springer, Cham, pp. 285–300. CrossRefGoogle Scholar
[16] Jin, P., Rüdiger, B. and Trabelsi, C. (2016). Positive Harris recurrence and exponential ergodicity of the basic affine jump-diffusion. Stoch. Anal. Appl. 34, 75–95. CrossRefGoogle Scholar
[17] Jin, P., Mandrekar, V., Rüdiger, B. and Trabelsi, C. (2013). Positive Harris recurrence of the CIR process and its applications. Commun. Stoch. Anal. 7, 409–424. Google Scholar
[18] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York. Google Scholar
[19] Keller-Ressel, M. (2011). Moment explosions and long-term behavior of affine stochastic volatility models. Math. Finance 21, 73–98. CrossRefGoogle Scholar
[20] Keller-Ressel, M. and Mijatović, A. (2012). On the limit distributions of continuous-state branching processes with immigration. Stoch. Process. Appl. 122, 2329–2345. CrossRefGoogle Scholar
[21] Keller-Ressel, M. and Steiner, T. (2008). Yield curve shapes and the asymptotic short rate distribution in affine one-factor models. Finance Stoch. 12, 149–172. Google Scholar
[22] Li, Z. and Ma, C. (2015). Asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model. Stoch. Process. Appl. 125, 3196–3233. CrossRefGoogle Scholar
[23] Long, H. (2010). Parameter estimation for a class of stochastic differential equations driven by small stable noises from discrete observations. Acta Math. Sci. B 30, 645–663. Google Scholar
[24] Meyn, S. P. and Tweedie, R. L. (1992). Stability of Markovian processes. I. Criteria for discrete-time chains. Adv. Appl. Prob. 24, 542–574. CrossRefGoogle Scholar
[25] 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
[26] 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
[27] Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press. CrossRefGoogle Scholar
[28] Overbeck, L. (1998). Estimation for continuous branching processes. Scand. J. Statist. 25, 111–126. Google Scholar
[29] Overbeck, L. and Rydén, T. (1997). Estimation in the Cox–Ingersoll–Ross model. Econometric Theory 13, 430–461. Google Scholar
[30] Sato, K.-I. (2013). Lévy Processes and Infinitely Divisible Distributions (Camb. Stud. Adv. Math. 68). Cambridge University Press. Google Scholar
[31] Situ, R. (2010). Theory of Stochastic Differential Equations with Jumps and Applications, Vol. 1. Springer, New York. Google Scholar
[32] Vasicek, O. (1977). An equilibrium characterization of the term structure. J. Financ. Econom. 5, 177–188. CrossRefGoogle Scholar