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Analysis of threshold Ornstein–Uhlenbeck process with piecewise linear drift and piecewise constant diffusion

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  • Zhang, Dingwen
  • Sun, Zhuowei

Abstract

The threshold Ornstein−Uhlenbeck process is a stochastic process defined by a stochastic differential equation, with a piecewise linear drift term and a piecewise constant diffusion term. We address the problem of estimating the drift and diffusion parameters along with the thresholds. To solve this problem, we propose two algorithms, LS&MQV&S and LS&QV&S, which incorporate the least squares method for the drift parameters, the modified quadratic variation and quadratic variation methods for the diffusion parameters, a testing procedure to determine the presence of additional drift- or diffusion-thresholds, and a sequential method for the values of the drift- and diffusion-thresholds. Monte Carlo simulation results are presented to illustrate and support our theoretical conclusions. We apply the process to model the U.S. Treasury rates and the currency foreign exchange rates, using the LS&QV&S algorithm to estimate the term structures of the drift and diffusion terms.

Suggested Citation

  • Zhang, Dingwen & Sun, Zhuowei, 2025. "Analysis of threshold Ornstein–Uhlenbeck process with piecewise linear drift and piecewise constant diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 199(P2).
    • Handle: RePEc:eee:chsofr:v:199:y:2025:i:p2:s0960077925006721
    DOI: 10.1016/j.chaos.2025.116659
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