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Approximating some Volterra type stochastic integrals with applications to parameter estimation

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  • Hult, Henrik

Abstract

We consider Volterra type processes which are Gaussian processes admitting representation as a Volterra type stochastic integral with respect to the standard Brownian motion, for instance the fractional Brownian motion. Gaussian processes can be represented as a limit of a sequence of processes in the associated reproducing kernel Hilbert space and as a special case of this representation, we derive Karhunen-Loéve expansions for Volterra type processes. In particular, a wavelet decomposition for the fractional Brownian motion is obtained. We also consider a Skorohod type stochastic integral with respect to a Volterra type process and using the Karhunen-Loéve expansions we show how it can be approximated. Finally, we apply the results to estimation of drift parameters in stochastic models driven by Volterra type processes using a Girsanov transformation and we prove consistency, the rate of convergence and asymptotic normality of the derived maximum likelihood estimators.

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  • Hult, Henrik, 2003. "Approximating some Volterra type stochastic integrals with applications to parameter estimation," Stochastic Processes and their Applications, Elsevier, vol. 105(1), pages 1-32, May.
    • Handle: RePEc:eee:spapps:v:105:y:2003:i:1:p:1-32
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    Cited by:

    1. Stefan Gerhold & Christoph Gerstenecker & Archil Gulisashvili, 2020. "Large deviations for fractional volatility models with non-Gaussian volatility driver," Papers 2003.12825, arXiv.org.
    2. Saussereau, Bruno & Stoica, Ion Lucretiu, 2012. "Scalar conservation laws with fractional stochastic forcing: Existence, uniqueness and invariant measure," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1456-1486.
    3. Gerhold, Stefan & Gerstenecker, Christoph & Gulisashvili, Archil, 2021. "Large deviations for fractional volatility models with non-Gaussian volatility driver," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 580-600.
    4. Miriana Cellupica & Barbara Pacchiarotti, 2021. "Pathwise Asymptotics for Volterra Type Stochastic Volatility Models," Journal of Theoretical Probability, Springer, vol. 34(2), pages 682-727, June.
    5. Čoupek, P. & Maslowski, B., 2017. "Stochastic evolution equations with Volterra noise," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 877-900.
    6. Archil Gulisashvili, 2018. "Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions," Papers 1808.00421, arXiv.org, revised Jun 2019.
    7. Dzhaparidze, Kacha & van Zanten, Harry & Zareba, Pawel, 2005. "Representations of fractional Brownian motion using vibrating strings," Stochastic Processes and their Applications, Elsevier, vol. 115(12), pages 1928-1953, December.
    8. Gulisashvili, Archil, 2020. "Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3648-3686.

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