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Geometric ergodicity of affine processes on cones

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  • Mayerhofer, Eberhard
  • Stelzer, Robert
  • Vestweber, Johanna

Abstract

For affine processes on finite-dimensional cones, we give criteria for geometric ergodicity — that is exponentially fast convergence to a unique stationary distribution. Ergodic results include both the existence of exponential moments of the limiting distribution, where we exploit the crucial affine property, and finite moments, where we invoke the polynomial property of affine semigroups. Furthermore, we elaborate sufficient conditions for aperiodicity and irreducibility. Our results are applicable to Wishart processes with jumps on the positive semidefinite matrices, continuous-time branching processes with immigration in high dimensions, and classical term-structure models for credit and interest rate risk.

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  • Mayerhofer, Eberhard & Stelzer, Robert & Vestweber, Johanna, 2020. "Geometric ergodicity of affine processes on cones," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4141-4173.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:7:p:4141-4173
    DOI: 10.1016/j.spa.2019.11.012
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    as
    1. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-1054, July.
    2. Chiarella, Carl & Da Fonseca, José & Grasselli, Martino, 2014. "Pricing range notes within Wishart affine models," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 193-203.
    3. Xiaowei Zhang & Peter W. Glynn, 2018. "Affine Jump-Diffusions: Stochastic Stability and Limit Theorems," Papers 1811.00122, arXiv.org.
    4. Christa Cuchiero & Josef Teichmann, 2011. "Path properties and regularity of affine processes on general state spaces," Papers 1107.1607, arXiv.org, revised Jan 2013.
    5. Mayerhofer, Eberhard & Muhle-Karbe, Johannes & Smirnov, Alexander G., 2011. "A characterization of the martingale property of exponentially affine processes," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 568-582, March.
    6. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    7. Boussama, Farid & Fuchs, Florian & Stelzer, Robert, 2011. "Stationarity and geometric ergodicity of BEKK multivariate GARCH models," Stochastic Processes and their Applications, Elsevier, vol. 121(10), pages 2331-2360, October.
    8. Branger, Nicole & Muck, Matthias & Seifried, Frank Thomas & Weisheit, Stefan, 2017. "Optimal portfolios when variances and covariances can jump," Journal of Economic Dynamics and Control, Elsevier, vol. 85(C), pages 59-89.
    9. Galtchouk, L. & Pergamenshchikov, S., 2007. "Uniform concentration inequality for ergodic diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 830-839, July.
    10. Qiang Dai & Kenneth J. Singleton, 2000. "Specification Analysis of Affine Term Structure Models," Journal of Finance, American Finance Association, vol. 55(5), pages 1943-1978, October.
    11. L. Galtchouk & S. Pergamenshchikov, 2011. "Adaptive sequential estimation for ergodic diffusion processes in quadratic metric," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(2), pages 255-285.
    12. Sato, Ken-iti & Yamazato, Makoto, 1984. "Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type," Stochastic Processes and their Applications, Elsevier, vol. 17(1), pages 73-100, May.
    13. Brockwell, Peter J. & Schlemm, Eckhard, 2013. "Parametric estimation of the driving Lévy process of multivariate CARMA processes from discrete observations," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 217-251.
    14. José da Fonseca & Martino Grasselli, 2011. "Riding on the smiles," Quantitative Finance, Taylor & Francis Journals, vol. 11(11), pages 1609-1632.
    15. Christa Cuchiero & Damir Filipovi'c & Eberhard Mayerhofer & Josef Teichmann, 2009. "Affine processes on positive semidefinite matrices," Papers 0910.0137, arXiv.org, revised Apr 2011.
    16. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    17. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    18. Alfonsi, Aurélien & Kebaier, Ahmed & Rey, Clément, 2016. "Maximum likelihood estimation for Wishart processes," Stochastic Processes and their Applications, Elsevier, vol. 126(11), pages 3243-3282.
    19. Mayerhofer, Eberhard, 2012. "Affine processes on positive semidefinite d×d matrices have jumps of finite variation in dimension d>1," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3445-3459.
    20. Mayerhofer, Eberhard & Pfaffel, Oliver & Stelzer, Robert, 2011. "On strong solutions for positive definite jump diffusions," Stochastic Processes and their Applications, Elsevier, vol. 121(9), pages 2072-2086, September.
    21. Gourieroux, Christian & Sufana, Razvan, 2010. "Derivative Pricing With Wishart Multivariate Stochastic Volatility," Journal of Business & Economic Statistics, American Statistical Association, vol. 28(3), pages 438-451.
    22. Martino Grasselli & Giulio Miglietta, 2016. "A flexible spot multiple-curve model," Quantitative Finance, Taylor & Francis Journals, vol. 16(10), pages 1465-1477, October.
    23. Christa Cuchiero & Martin Keller-Ressel & Josef Teichmann, 2012. "Polynomial processes and their applications to mathematical finance," Finance and Stochastics, Springer, vol. 16(4), pages 711-740, October.
    24. S. Haug & C. Klüppelberg & A. Lindner & M. Zapp, 2007. "Method of moment estimation in the COGARCH(1,1) model," Econometrics Journal, Royal Economic Society, vol. 10(2), pages 320-341, July.
    25. Aurélien Alfonsi, 2015. "Affine Diffusions and Related Processes: Simulation, Theory and Applications," Post-Print hal-03127212, HAL.
    26. Johannes Muhle-Karbe & Oliver Pfaffel & Robert Stelzer, 2010. "Option Pricing in Multivariate Stochastic Volatility Models of OU Type," Papers 1001.3223, arXiv.org, revised Nov 2011.
    27. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
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