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Bayesian inference for partially observed stochastic differential equations driven by fractional Brownian motion

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Listed:
  • A. Beskos
  • J. Dureau
  • K. Kalogeropoulos

Abstract

We consider continuous-time diffusion models driven by fractional Brownian motion. Observations are assumed to possess a nontrivial likelihood given the latent path. Due to the non-Markovian and high-dimensional nature of the latent path, estimating posterior expectations is computationally challenging. We present a reparameterization framework based on the Davies and Harte method for sampling stationary Gaussian processes and use it to construct a Markov chain Monte Carlo algorithm that allows computationally efficient Bayesian inference. The algorithm is based on a version of hybrid Monte Carlo simulation that delivers increased efficiency when used on the high-dimensional latent variables arising in this context. We specify the methodology on a stochastic volatility model, allowing for memory in the volatility increments through a fractional specification. The method is demonstrated on simulated data and on the S&P 500/VIX time series. In the latter case, the posterior distribution favours values of the Hurst parameter smaller than $1/2$, pointing towards medium-range dependence.

Suggested Citation

  • A. Beskos & J. Dureau & K. Kalogeropoulos, 2015. "Bayesian inference for partially observed stochastic differential equations driven by fractional Brownian motion," Biometrika, Biometrika Trust, vol. 102(4), pages 809-827.
  • Handle: RePEc:oup:biomet:v:102:y:2015:i:4:p:809-827.
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    File URL: http://hdl.handle.net/10.1093/biomet/asv051
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    References listed on IDEAS

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    1. Lobato, Ignacio N & Savin, N E, 1998. "Real and Spurious Long-Memory Properties of Stock-Market Data," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(3), pages 261-268, July.
    2. Lobato, Ignacio N & Savin, N E, 1998. "Real and Spurious Long-Memory Properties of Stock-Market Data: Reply," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(3), pages 280-283, July.
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    Cited by:

    1. Paramahansa Pramanik & Edward L. Boone & Ryad A. Ghanam, 2024. "Parametric Estimation in Fractional Stochastic Differential Equation," Stats, MDPI, vol. 7(3), pages 1-16, July.
    2. Elisa Alòs & Maria Elvira Mancino & Tai-Ho Wang, 2019. "Volatility and volatility-linked derivatives: estimation, modeling, and pricing," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 321-349, December.
    3. Qi Zhao & Alexandra Chronopoulou, 2023. "Delta-hedging in fractional volatility models," Annals of Finance, Springer, vol. 19(1), pages 119-140, March.

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