The thesis is devoted to the study of fractional models of financial mathematics and the solution of related statistical problems. In particular, the fractional Vasicek model and the mixed fractional Brownian motion with trend were investigated.
Nature is full of various phenomena which can be represented as processes with random evolution through time. Traditionally a standard Brownian motion is used to mathematically model such time series. However, different studies have shown that some processes exhibit self-similarity, long-range dependence and complex correlation structures. Usage of fractional Brownian motion allows to model such processes, since it has correlated increments which imply short-range dependence for Hurst index less than 1/2 and long-range dependence for Hurst index greater than 1/2.
Probabilistic model, proposed in 1977 by O. Vasicek for modeling interest rates, is widely used not only in economics and financial mathematics, but also in many other fields. To model processes with long-range dependence property that arise in finance, economics, hydrology and telecommunications, a generalization of this model, fractional Vasicek model, was proposed. It was studied in P. Cheridito et al. (2003), F. Comte et al. (2012), W. Xiao et al. (2014) and others.
Currently, theory of parametric estimation is well developed in the partial case of the model with one unknown parameter. See e. g. M. Kleptsyna and A. Le Breton (2002), Y. Mishura (2008), Y. Hu and D. Nualart (2010), K. Es-Sebaiy (2013), K. Tanaka (2013), K. Kubilius et al. (2015), Y. Kozachenko et al. (2015), K. Kubilius et al. (2017), A. Kukush et al. (2017).
However, in applied problems there is a need for a more flexible two-parameter model. Relevant studies were conducted by Y. Kutoyants (2004), but for a classical Vasicek model driven by Wiener process.
In the thesis the study of estimation problem for the fractional Vasicek model with two unknown parameters is considered. For the fractional Vasicek model so called ergodic type estimators are constructed for both continuous and discrete trajectories of the process. In both cases strong consistency of estimators is proved. Numerical simulation of the ergodic type estimators is performed for the discrete case.
In addition, maximum likelihood estimators of unknown parameters are constructed in the case of continuous observations: the estimators of each parameter when another one is assumed to be known, and the estimator of vector parameter for simultaneous estimation. For considered estimators consistency is proved and their asymptotic distributions are found. The important fact of asymptotic independence of maximum likelihood estimators for two unknown parameters is established.
The thesis is also devoted to the study of the mixed fractional Brownian motion with trend. This model, which was introduced in P. Cheridito (2001), found its applications, e.g., in computer network traffic modelling and more widely in finance.
Parameter estimation problem in the mixed fractional Brownian motion without trend was studied e. g. in M. Dozzi et al. (2015), D. Filatova (2008), W.-L. Xiao et al. (2011), P. Zhang et al. (2014).
C. Cai et al. (2012) considered estimation of the drift parameter assuming that Hurst index and coefficient by the Wiener process are known and coefficient by the fractional Brownian motion equals 1. Notice that this approach requires to know the solution to an integral equation. Hence, it is difficult to discretize the estimator, and especially, to adapt it to the case of unknown Hurst index and coefficient by the Wiener process.
To the best of our knowledge, simultaneous estimation of all four parameters of the mixed fractional Brownian motion with trend was studied only in J. Dufitinema et al. (2020), but with a slightly different parametrization.
Estimation of drift parameter in similar models with more general noises was studied in papers Y. Mishura et al. (2015–2018).
In the thesis two approaches to simultaneous estimation of all four parameters of the mixed fractional Brownian motion with trend are investigated.
The first algorithm is more traditional. First, strong consistency and asymptotic normality of known estimator of the drift parameter are proved. Then, it’s components are replaced by strongly consistent estimators of other parameters, which are based on quadratic variations. Strong consistency of obtained plug-in estimator is proved. However, this approach has several limitations. Therefore, a new approach based on the ergodic theorem is developed. It allows to estimate simultaneously all unknown parameters under much more general conditions. In the thesis, strong consistency of constructed estimators is proved. Also asymptotic normality of the estimator of the drift parameter is established. Finally, effectiveness of both estimators of the drift parameter is compared.
Results of numerical simulations for all estimators constructed by two methods are shown.
