On estimation of the Hurst index of solutions of stochastic differential equations

Dmitrij Melichov

Doctoral dissertation

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Physical Sciences, Mathematics (01P).

The main topic of this dissertation is the estimation of the Hurst index H of the solutions of stochastic differential equations (SDEs) driven by the fractional Brownian motion (fBm).

Firstly, the limit behavior of the first and second order quadratic variations of the solutions of SDEs driven by the fBm is analyzed. This yields several strongly consistent estimators of the Hurst index H. Secondly, it is proved that in case the solution of the SDE is replaced by its Milstein approximation, the estimators remain strongly consistent. Additionally, the possibilities of applying the increment ratios (IR) statistic based estimator of H originally obtained by J. M. Bardet and D. Surgailis in 2010 to the fractional geometric Brownian motion are examined. Furthermore, this dissertation derives the convergence rate of the modified Gladyshev’s estimator of the Hurst index to its real value.

The estimators obtained in the dissertation were compared with several other known estimators of the Hurst index H, namely the naive and ordinary least squares Gladyshev and _-summing oscillation estimators, the variogram estimator and the IR estimator. The models chosen for comparison of these estimators were the fractional Ornstein-Uhlenbeck (O-U) process and the fractional geometric Brownian motion (gBm). The initial inference about the behavior of these estimators was drawn for the O-U process which is Gaussian, while the gBm process was used to check how the estimators behave in a non- Gaussian case. The scope of modelling was 100 sample paths of the length n = 214 + 1 for each value of H 2 {0.55, 0.6, . . . , 0.95} on the unit interval t 2 [0, 1].

The dissertation consists of the introduction, 3 main chapters, the conclusions, the bibliography, the list of author’s publications on the topic of dissertation and two appendices.

The results obtained during the doctoral studies were published in 6 papers in reviewed periodic scientific journals and were presented at 5 conferences, of which 2 – international.

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DOI: https://doi.org/10.20334/1923-M

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Imprint No:
160×230 mm
90 p.
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