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

Part of: Ergodic theory Markov processes

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.
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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 

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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