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Parameter estimation for reflected Ornstein–Uhlenbeck processes with discrete observations

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Abstract

A parameter estimation problem for a one-dimensional reflected Ornstein–Uhlenbeck is considered. We assume that only the state process itself (not the local time process) is observable and the observations are made only at discrete time instants. Strong consistency and asymptotic normality are established. Our approach is of the method of moments type and is based on the explicit form of the invariant density of the process. The method is valid irrespective of the length of the time intervals between consecutive observations.

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References

  • Attia FA (1991) On a reflected Ornstein–Uhlenbeck process with an application. Bull Aust Math Soc 43(3):519–528

    Article  MATH  MathSciNet  Google Scholar 

  • Bo L, Wang Y, Yang X (2010) Some integral functionals of reflected SDEs and their applications in finance. Quant Financ 11(3):343–348

    Article  MathSciNet  Google Scholar 

  • Bo L, Tang D, Wang Y, Yang X (2011a) On the conditional default probability in a regulated market: a structural approach. Quant Financ 11(12):1695–1702

    Article  MATH  MathSciNet  Google Scholar 

  • Bo L, Wang Y, Yang X, Zhang G (2011b) Maximum likelihood estimation for reflected Ornstein–Uhlenbeck processes. J Stat Plan Inference 141(1):588–596

    Article  MATH  MathSciNet  Google Scholar 

  • Brooks S, Gelman A, Jones G, Meng XL (2011) Handbook of Markov Chain Monte Carlo. CRC Press, Boca Raton

    Book  MATH  Google Scholar 

  • Budhiraja A, Lee C (2007) Long time asymptotics for constrained diffusions in polyhedral domains. Stoch Process Appl 117(8):1014–1036

    Article  MATH  MathSciNet  Google Scholar 

  • Glynn PW, Meyn SP (1996) A Liapounov bound for solutions of the Poisson equation. Ann Appl Probab 24(2):916–931

    Article  MATH  MathSciNet  Google Scholar 

  • Harrison JM (1985) Brownian motion and stochastic flow systems. Wiley series in probability and mathematical statistics. Wiley, New York

    Google Scholar 

  • Hu Y, Song J (2013) Parameter estimation for fractional Ornstein–Uhlenbeck processes with discrete observations. Malliavin calculus and stochastic analysis. Springer, US, pp 427–442

  • Lee C, Bishwal JPN, Lee MH (2012) Sequential maximum likelihood estimation for reflected Ornstein–Uhlenbeck processes. J Stat Plan Inference 142(5):1234–1242

    Article  MATH  MathSciNet  Google Scholar 

  • Lépingle D (1995) Euler scheme for reflected stochastic differential equations. Math Comput Simul 38(1–3):119–126

    Article  MATH  Google Scholar 

  • Linetsky V (2005) On the transition densities for reflected diffusions. Adv Appl Probab 37(2):435–460

    Article  MATH  MathSciNet  Google Scholar 

  • Lions PL, Sznitman AS (1984) Stochastic differential equations with reflecting boundary conditions. Commun Pure Appl Math 37(4):511–537

    Article  MATH  MathSciNet  Google Scholar 

  • Meyn SP, Tweedie RL (1993) Stability of Markovian processes II: continous-time processes and sampled chains. Adv Appl Probab 25:497–517

    Google Scholar 

  • Meyn SP, Tweedie RL (2009) Markov chains and stochastic stability, 2nd edn. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Ricciardi LM (1985) Stochastic population models II. Diffusion models. Lecture Notes at the International School on Mathematical Ecology

  • van der Vaart AW (1998) Asymptotic statistics. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press, Cambridge

    Google Scholar 

  • Ward AR, Glynn PW (2003a) A diffusion approximation for a Markovian queue with reneging. Queueing Syst 43(1–2):103–128

    Article  MATH  MathSciNet  Google Scholar 

  • Ward AR, Glynn PW (2003b) Properties of the reflected Ornstein–Uhlenbeck process. Queueing Syst 44(2):109–123

    Article  MATH  MathSciNet  Google Scholar 

  • Ward AR, Glynn PW (2005) A diffusion approximation for a \(GI/GI/1\) queue with balking or reneging. Queueing Syst 50(4):371–400

    Article  MATH  MathSciNet  Google Scholar 

  • Whitt W (2002) Stochastic-process limits. Springer series in operations research. Springer, New York

    Google Scholar 

  • Xing X, Zhang W, Wang Y (2009) The stationary distributions of two classes of reflected Ornstein–Uhlenbeck processes. J Appl Probab 46:709–720

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The research work of Y. Hu is supported in part by the Simons Foundation Grant (209206) and a General Research Fund of University of Kansas, and C. Lee is supported in part by the Simons Foundation Grant (209658) and the Army Research Office (W911NF-14-1-0216). The authors would like to thank the anonymous reviewers for their valuable comments and suggestions that have much enhanced the quality of the paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    Yaozhong Hu

  2. Department of Statistics, Colorado State University, Fort Collins, CO, 80523, USA

    Chihoon Lee & Myung Hee Lee

  3. Department of Mathematics and Department of Statistics and Actuarial Science, University of Hong Kong, Hong Kong, China

    Jian Song

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  1. Yaozhong Hu
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  2. Chihoon Lee
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  3. Myung Hee Lee
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  4. Jian Song
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Correspondence to Chihoon Lee.

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Hu, Y., Lee, C., Lee, M.H. et al. Parameter estimation for reflected Ornstein–Uhlenbeck processes with discrete observations. Stat Inference Stoch Process 18, 279–291 (2015). https://doi.org/10.1007/s11203-014-9112-7

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  • DOI: https://doi.org/10.1007/s11203-014-9112-7

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