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Journal of Theoretical Probability

, Volume 32, Issue 1, pp 183–201 | Cite as

Asymptotic Behaviour of the Trajectory Fitting Estimator for Reflected Ornstein–Uhlenbeck Processes

  • Qingpei ZangEmail author
  • Lixin Zhang
Article

Abstract

The Ornstein–Uhlenbeck process with reflection, which has been the subject of an enormous body of literature, both theoretical and applied, is a process that returns continuously and immediately to the interior of the state space when it attains a certain boundary. In this work, we are mainly concerned with the study of the asymptotic behavior of the trajectory fitting estimator for nonergodic reflected Ornstein–Uhlenbeck processes, including strong consistency and asymptotic distribution. Moreover, we also prove that this kind of estimator for ergodic reflected Ornstein–Uhlenbeck processes does not possess the property of strong consistency.

Keywords

Reflected Ornstein–Uhlenbeck processes Trajectory fitting estimator Nonergodic 

Mathematics Subject Classification (2010)

Primary 60F15 Secondary 62F12 

Notes

Acknowledgements

The authors are grateful to the referees and associate editor for constructive comments which led to improvement of this work. We also thank Professor Hui Jiang at Nanjing University of Aeronautics and Astronautics for his comments on the revised version. This work was completed when the first author was visiting the University of Kansas in 2015; this author would like to thank Professor Yaozhong Hu in the Department of Mathematics at the University of Kansas for his warm hospitality. Zang acknowledges partial research support from National Natural Science Foundation of China (Grant Nos. 11326174 and 11401245), Natural Science Foundation of Jiangsu Province (Grant No. BK20130412), Natural Science Research Project of Ordinary Universities in Jiangsu Province (Grant No. 12KJB110003), China Postdoctoral Science Foundation (Grant No. 2014M551720), Jiangsu Government Scholarship for Overseas Studies, and Qing Lan project of Jiangsu Province (2016). Zhang acknowledges partial research support Zhang acknowledges partial research support from National Natural Science Foundation of China (Grant No. 11225104 and 11731012), Zhejiang Provincial Natural Science Foundation (Grant No. R6100119) and the Fundamental Research Funds for the Central Universities.

References

  1. 1.
    Asmussen, S., Avram, F., Pistorius, M.R.: Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79–111 (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Asmussen, S., Pihlsgard, M.: Loss rates for Lévy processes with two reflecting barriers. Math. Methods Oper. Res. 32, 308–321 (2007)CrossRefzbMATHGoogle Scholar
  3. 3.
    Ata, B., Harrison, J.M., Shepp, L.A.: Drift rate control of a Brownian processing system. Ann. Appl. Probab. 15, 1145–1160 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Atar, R., Budhiraja, A.: Stability properties of constrained jump-diffusion processes. Electron. J. Probab. 7, 1–31 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Avram, F., Kyprianou, A.E., Pistorius, M.R.: Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Probab. 14, 215–238 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Avram, F., Palmowski, Z., Pistorius, M.R.: On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Probab. 17, 156–180 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bishwal, J.P.N.: Parameter Estimation in Stochastic Differential Equations. Lecture Notes in Mathematics, vol. 1923. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bo, L., Wang, Y., Yang, X.: Some integral functionals of reflected SDEs and their applications in finance. Quant. Finance 11(3), 343–348 (2011a)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bo, L., Tang, D., Wang, Y., Yang, X.: On the conditional default probability in a regulated market: a structural approach. Quant. Finance 11(12), 1695–1702 (2011b)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bo, L., Wang, Y., Yang, X., Zhang, G.: Maximum likelihood estimation for reflected Ornstein–Uhlenbeck processes. J. Stat. Plan. Inference 141, 588–596 (2011c)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bo, L., Wang, Y., Yang, X.: First passage times of (reflected) Ornstein–Uhlenbeck processes over random jump boundaries. J. Appl. Probab. 48(3), 723–732 (2011d)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bo, L., Ren, G., Wang, Y., Yang, X.: First passage times of reflected generalized Ornstein–Uhlenbeck processes. Stoch. Dyn. 13, 1–16 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bo, L., Yang, X.: Sequential maximum likelihood estimation for reflected generalized Ornstein–Uhlenbeck processes. Stat. Probab. Lett. 82, 1374–1382 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dietz, H.M.: Asymptotic behavior of trajectory fitting estimators for certain non-ergodic SDE. Stat. Inference Stoch. Process. 4, 249–258 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dietz, H.M., Kutoyants, Y.A.: A class of minimum-distance estimators for diffusion processes with ergodic properties. Stat. Decis. 15, 211–227 (1997)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Dietz, H.M., Kutoyants, Y.A.: Parameter estimation for some non-recurrent solutions of SDE. Stat. Decis. 21, 29–45 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Feigin, P.D.: Some comments concerning a curious singularity. J. Appl. Probab. 16, 440–444 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Goldstein, R.S., Keirstead, W.P.: On the term structure of interest rates in the presence of reflecting and absorbing boundaries (1997).  https://doi.org/10.2139/ssrn.10.2139/ssrn.19840
  19. 19.
    Hanson, S.D., Myers, R.J., Hilker, J.H.: Hedging with futures and options under truncated cash price distribution. J. Agric. Appl. Econ. 31, 449–459 (1999)CrossRefGoogle Scholar
  20. 20.
    Harrison, M.: Brownian Motion and Stochastic Flow Systems. Wiley, New York (1986)Google Scholar
  21. 21.
    Hu, Y., Long, H.W.: Parameter estimation for Ornstein–Uhlenbeck processes drvien by \(\alpha \)stable lévy motions. Commun. Stoch. Anal. 1(2), 175–192 (2007)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Hu, Y., Lee, C.: Parameter estimation for a reflected fractional Brownian motion based on its local time. J. Appl. Probab. 50, 592–597 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hu, Y., Lee, C., Lee, M.H., Song, J.: Parameter estimation for reflected Ornstein–Uhlenbeck processes with discrete observations Stat. Inference Stoch. Process. 18, 279–291 (2015)Google Scholar
  24. 24.
    Jiang, H., Xie, C.: Asymptotic behaviours for the trajectory fitting estimator in Ornstein–Uhlenbeck process with linear drift. Stochastics 88, 336–352 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1991)zbMATHGoogle Scholar
  26. 26.
    Krugman, P.R.: Target zones and exchange rate dynamics. Quart. J. Econ. 106, 669–682 (1991)CrossRefGoogle Scholar
  27. 27.
    Kutoyants, Y.A.: Minimum distance parameter estimation for diffusion type observations. Comptes Rendus de l’Académie des Sciences, Series I 312, 637–642 (1991)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Kutoyants, Y.A.: Statiatical Inference for Ergodic Diffusion Processes. Springer, London (2004)CrossRefzbMATHGoogle Scholar
  29. 29.
    Lee, C., Bishwal, J.P.N., Lee, M.H.: Sequential maximum likelihood estimation for reflected Ornstein–Uhlenbeck processes. J. Stat. Plan. Inference 142, 1234–1242 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lee, C., Song, J.: On drift parameter estimation for reflected fractional Ornstein–Uhlenbeck processes (2013). arXiv:1303.6379
  31. 31.
    Linetsky, V.: On the transition densities for reflected diffusions. Adv. Appl. Probab. 37, 435–460 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Lions, P.L., Sznitman, A.S.: Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math. 37, 511–537 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Major, P.: Tail behaviour of multiple random integrals and U-statistics. Probab. Surv. 2, 448–505 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Major, P.: On a multivariate version of Bernsteins inequality. Electron. J. Probab. 12, 966–988 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Mandjes, M., Spreij, P.: A note on the central limit theorem for a one-sided reflected Ornstein–Uhlenbeck process (2016). arXiv:1601.05653v1
  36. 36.
    Mao, X.R.: Stochastic Differential Equations and Applications. Horwood Publishing, Chichester (2007)zbMATHGoogle Scholar
  37. 37.
    Prakasa Rao, B.L.S.: Statistical Inference for Diffusion Type Processes. Oxford University Press, New York (1999)zbMATHGoogle Scholar
  38. 38.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1998)zbMATHGoogle Scholar
  39. 39.
    Ricciardi, L.M., Sacerdote, L.: On the probability densities of an Ornstein–Uhlenbeck process with a reflecting boundary. J. Appl. Probab. 24, 355–369 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ward, A., Glynn, P.: A diffusion approximation for Markovian queue with reneging. Queueing Syst. 43, 103–128 (2003a)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Ward, A., Glynn, P.: Properties of the reflected Ornstein–Uhlenbeck process. Queueing Syst. 44, 109–123 (2003b)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Ward, A., Glynn, P.: A diffusion approximation for a GI/GI/1 queue with balking or reneging. Queueing Syst. 50(4), 371–400 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Ward, W.: Stochastic-Process Limits. Springer Series in Operations Research. Springer, New York (2002)Google Scholar
  44. 44.
    Xing, X., Zhang, W., Wang, Y.: The stationary distributions of two classes of reflected Ornstein–Uhlenbeck processes. J. Appl. Probab. 46, 709–720 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Zang, Q., Zhang, L.: Parameter estimation for generalized diffusion processes with reflected boundary. Sci. China Math. 59(6), 1163–1174 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mathematical ScienceHuaiyin Normal UniversityHuai’anPeople’s Republic of China
  2. 2.School of Mathematical SciencesZhejiang UniversityHangzhouPeople’s Republic of China

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