The Annals of Statistics

Statistical aspects of the fractional stochastic calculus

Ciprian A. Tudor and Frederi G. Viens

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We apply the techniques of stochastic integration with respect to fractional Brownian motion and the theory of regularity and supremum estimation for stochastic processes to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equations driven by a fractional Brownian motion with any level of Hölder-regularity (any Hurst parameter). We prove existence and strong consistency of the MLE for linear and nonlinear equations. We also prove that a version of the MLE using only discrete observations is still a strongly consistent estimator.

Article information

Ann. Statist., Volume 35, Number 3 (2007), 1183-1212.

First available in Project Euclid: 24 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M09: Non-Markovian processes: estimation
Secondary: 60G18: Self-similar processes 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05]

Maximum likelihood estimator fractional Brownian motion strong consistency stochastic differential equation Malliavin calculus Hurst parameter


Tudor, Ciprian A.; Viens, Frederi G. Statistical aspects of the fractional stochastic calculus. Ann. Statist. 35 (2007), no. 3, 1183--1212. doi:10.1214/009053606000001541.

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