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Large deviations for statistics of the Jacobi process

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  • Demni, N.
  • Zani, M.

Abstract

This paper aims to derive large deviations for statistics of the Jacobi process already conjectured by M. Zani in her thesis. To proceed, we write in a simpler way the Jacobi semi-group density. Being given by a bilinear sum involving Jacobi polynomials, it differs from Hermite and Laguerre cases by the quadratic form of its eigenvalues. Our attempt relies on subordinating the process using a suitable random time change. This gives a Mehler-type formula whence we recover the desired semi-group density. Once we do, an adaptation of Zani's result [M. Zani, Large deviations for squared radial Ornstein-Uhlenbeck processes, Stochastic. Process. Appl. 102 (1) (2002) 25-42] to the non-steep case will provide the required large deviations principle.

Suggested Citation

  • Demni, N. & Zani, M., 2009. "Large deviations for statistics of the Jacobi process," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 518-533, February.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:2:p:518-533
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    References listed on IDEAS

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    1. Zani, Marguerite, 2002. "Large deviations for squared radial Ornstein-Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 102(1), pages 25-42, November.
    2. Ludger Overbeck, 1998. "Estimation for Continuous Branching Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 25(1), pages 111-126, March.
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    Cited by:

    1. Damien Ackerer & Damir Filipovic & Sergio Pulido, 2017. "The Jacobi Stochastic Volatility Model," Working Papers hal-01338330, HAL.
    2. Yun, Youngyun, 2018. "The moments of a diffusion process," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 36-41.
    3. Bercu, Bernard & Richou, Adrien, 2017. "Large deviations for the Ornstein–Uhlenbeck process without tears," Statistics & Probability Letters, Elsevier, vol. 123(C), pages 45-55.
    4. László Márkus & Ashish Kumar, 2021. "Modelling Joint Behaviour of Asset Prices Using Stochastic Correlation," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 341-354, March.
    5. Zhao, Shoujiang & Gao, Fuqing, 2010. "Large deviations in testing Jacobi model," Statistics & Probability Letters, Elsevier, vol. 80(1), pages 34-41, January.
    6. Damien Ackerer & Damir Filipović & Sergio Pulido, 2018. "The Jacobi stochastic volatility model," Finance and Stochastics, Springer, vol. 22(3), pages 667-700, July.
    7. Damir Filipovic & Damien Ackerer & Sergio Pulido, 2018. "The Jacobi Stochastic Volatility Model," Post-Print hal-01338330, HAL.
    8. Damien Ackerer & Damir Filipovi'c & Sergio Pulido, 2016. "The Jacobi Stochastic Volatility Model," Papers 1605.07099, arXiv.org, revised Mar 2018.

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