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A Hoeffding’s inequality for uniformly ergodic diffusion process

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  • Choi, Michael C.H.
  • Li, Evelyn

Abstract

In this note, we present a version of Hoeffding’s inequality in a continuous-time setting, where the data stream comes from a uniformly ergodic diffusion process. Similar to the well-studied case of Hoeffding’s inequality for discrete-time uniformly ergodic Markov chain, the proof relies on techniques ranging from martingale theory to classical Hoeffding’s lemma as well as the notion of deviation kernel of diffusion process. We present two examples to illustrate our results. In the first example we consider large deviation probability on the occupation time of the Jacobi diffusion, a popular process used in modelling of exchange rates in mathematical finance, while in the second example we look at the exponential functional of a finite interval analogue of the Ornstein–Uhlenbeck process introduced by Kessler and Sørensen (1999).

Suggested Citation

  • Choi, Michael C.H. & Li, Evelyn, 2019. "A Hoeffding’s inequality for uniformly ergodic diffusion process," Statistics & Probability Letters, Elsevier, vol. 150(C), pages 23-28.
    • Handle: RePEc:eee:stapro:v:150:y:2019:i:c:p:23-28
    DOI: 10.1016/j.spl.2019.02.012
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    1. Julie Lyng Forman & Michael Sørensen, 2008. "The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(3), pages 438-465, September.
    2. Ward Whitt, 1992. "Asymptotic Formulas for Markov Processes with Applications to Simulation," Operations Research, INFORMS, vol. 40(2), pages 279-291, April.
    3. 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.
    4. Glynn, Peter W. & Ormoneit, Dirk, 2002. "Hoeffding's inequality for uniformly ergodic Markov chains," Statistics & Probability Letters, Elsevier, vol. 56(2), pages 143-146, January.
    5. Boucher, Thomas R., 2009. "A Hoeffding inequality for Markov chains using a generalized inverse," Statistics & Probability Letters, Elsevier, vol. 79(8), pages 1105-1107, April.
    6. Kristian Stegenborg Larsen & Michael Sørensen, 2007. "Diffusion Models For Exchange Rates In A Target Zone," Mathematical Finance, Wiley Blackwell, vol. 17(2), pages 285-306, April.
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    Cited by:

    1. Sandrić, Nikola & Šebek, Stjepan, 2023. "Hoeffding’s inequality for non-irreducible Markov models," Statistics & Probability Letters, Elsevier, vol. 200(C).
    2. Liu, Jinpeng & Liu, Yuanyuan & Zhao, Yiqiang Q., 2022. "Augmented truncation approximations to the solution of Poisson’s equation for Markov chains," Applied Mathematics and Computation, Elsevier, vol. 414(C).

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