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Limit theorems for reflected Ornstein–Uhlenbeck processes

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  • Gang Huang
  • Michel Mandjes
  • Peter Spreij

Abstract

type="main"> This paper studies one-dimensional Ornstein–Uhlenbeck (OU) processes, with the distinguishing feature that they are reflected on a single boundary (put at level 0) or two boundaries (put at levels 0 and d > 0). In the literature, they are referred to as reflected OU (ROU) and doubly reflected OU (DROU), respectively. For both cases, we explicitly determine the decay rates of the (transient) probability to reach a given extreme level. The methodology relies on sample-path large deviations, so that we also identify the associated most likely paths. For DROU, we also consider the ‘idleness process’ L t and the ‘loss process’ U t , which are the minimal non-decreasing processes, which make the OU process remain ≥ 0 and ≤ d, respectively. We derive central limit theorems (CLTs) for U t and L t , using techniques from stochastic integration and the martingale CLT.

Suggested Citation

  • Gang Huang & Michel Mandjes & Peter Spreij, 2014. "Limit theorems for reflected Ornstein–Uhlenbeck processes," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 68(1), pages 25-42, February.
    • Handle: RePEc:bla:stanee:v:68:y:2014:i:1:p:25-42
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    File URL: http://hdl.handle.net/10.1111/stan.12021
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    Cited by:

    1. Archil Gulisashvili, 2020. "Large deviation principles for stochastic volatility models with reflection and three faces of the Stein and Stein model," Papers 2006.15431, arXiv.org.
    2. Michel Mandjes & Peter Spreij, 2017. "A note on the central limit theorem for the idleness process in a one-sided reflected Ornstein–Uhlenbeck model," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 71(3), pages 225-235, August.
    3. Hui Jiang & Qingshan Yang, 2022. "Moderate Deviations for Drift Parameter Estimations in Reflected Ornstein–Uhlenbeck Process," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1262-1283, June.

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