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Model verification for Lévy-driven Ornstein–Uhlenbeck processes with estimated parameters

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  • Abdelrazeq, Ibrahim

Abstract

When an Ornstein–Uhlenbeck (or CAR(1)) process is observed at discrete times 0, h, 2h, …[T/h]h, the unobserved driving process can be approximated from the observed process. Approximated increments of the driving process are used to test the assumption that the process is Lévy-driven. Asymptotic behavior of the test statistic at high sampling frequencies is developed in Abdelrazeq et al. (2014) assuming that the model parameters a,σ are known. Here we explore the performance of the test statistic when the model coefficient a is unknown and must be estimated. The parameter σ can be assumed to be one. We will show the consistency and asymptotic normality of our proposed estimator and then demonstrate its effect on the asymptotic behavior of the test statistic. Performance of the proposed test with estimated a is illustrated through simulation.

Suggested Citation

  • Abdelrazeq, Ibrahim, 2015. "Model verification for Lévy-driven Ornstein–Uhlenbeck processes with estimated parameters," Statistics & Probability Letters, Elsevier, vol. 104(C), pages 26-35.
    • Handle: RePEc:eee:stapro:v:104:y:2015:i:c:p:26-35
    DOI: 10.1016/j.spl.2015.04.014
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    1. Prakasa Rao, B. L. S., 1984. "On the exponential rate of convergence of the least squares estimator in the nonlinear regression model with Gaussian errors," Statistics & Probability Letters, Elsevier, vol. 2(3), pages 139-142, May.
    2. Prakasa Rao, B. L. S., 1984. "The rate of convergence of the least squares estimator in a non-linear regression model with dependent errors," Journal of Multivariate Analysis, Elsevier, vol. 14(3), pages 315-322, June.
    3. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
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    1. Gong, Xiao-li & Zhuang, Xin-tian, 2016. "Option pricing and hedging for optimized Lévy driven stochastic volatility models," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 118-127.
    2. Gong, Xiaoli & Zhuang, Xintian, 2016. "Option pricing for stochastic volatility model with infinite activity Lévy jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 455(C), pages 1-10.

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