Traditional financial modeling is based on semimartingale processes with stationary and independent increments. However, empirical investigations on financial data does not always support these assumptions. This contradiction showed that there is a need for new stochastic models. Fractional Brownian motion (fBm) was proposed as one of these models by Benoit Mandelbrot. FBm is the only continuous Gaussian process with dependent increments. Correlation between increments of a fBm changes according to its self-similarity parameter H. This property of fBm helps to capture the correlation dynamics of the data and consequently obtain better forecast results. But for values of H different than 1/2, fBm is not a semimartingale and classical Ito formula does not exist in that case. This gives rise to need for using the white noise theory to construct integrals with respect to fBm and obtain fractional Ito formulas. In this thesis, the representation of fBm and its fundamental properties are examined. Construction of Wick-Ito-Skorohod (WIS) and fractional WIS integrals are investigated. An Ito type formula and Girsanov type theorems are stated. The financial applications of fBm are mentioned and the Black&amp / Scholes price of a European call option on an asset which is assumed to follow a geometric fBm is derived. The statistical aspects of fBm are investigated. Estimators for the self-similarity parameter H and simulation methods of fBm are summarized. Using the R/S methodology of Hurst, the estimations of the parameter H are obtained and these values are used to evaluate the fractional Black&amp / Scholes prices of a European call option with different maturities. Afterwards, these values are compared to Black&amp / Scholes price of the same option to demonstrate the effect of long-range dependence on the option prices. Also, estimations of H at different time scales are obtained to investigate the multiscaling in financial data. An outlook of the future work is given.

<p>The first part of this thesis studies tail probabilities forelliptical distributions and probabilities of extreme eventsfor multivariate stochastic processes. It is assumed that thetails of the probability distributions satisfy a regularvariation condition. This means, roughly speaking, that thereis a non-negligible probability for very large or extremeoutcomes to occur. Such models are useful in applicationsincluding insurance, finance and telecommunications networks.It is shown how regular variation of the marginals, or theincrements, of a stochastic process implies regular variationof functionals of the process. Moreover, the associated tailbehavior in terms of a limit measure is derived.</p><p>The second part of the thesis studies problems related toparameter estimation in stochastic models with long memory.Emphasis is on the estimation of the drift parameter in somestochastic differential equations driven by the fractionalBrownian motion or more generally Volterra-type processes.Observing the process continuously, the maximum likelihoodestimator is derived using a Girsanov transformation. In thecase of discrete observations the study is carried out for theparticular case of the fractional Ornstein-Uhlenbeck process.For this model Whittle’s approach is applied to derive anestimator for all unknown parameters.</p>

In the thesis, multivariate fractional Brownian motions with possibly different Hurst indices in different coordinates are considered and a Girsanov-type theo- rem for these processes is shown. Two applications of this theorem to stochastic differential equations driven by multivariate fractional Brownian motions (SDEs) are given. Firstly, the existence of a weak solution to an SDE with a drift coeffi- cient that can be written as a sum of a regular and a singular part and a diffusion coefficient that is dependent on time and satisfies suitable conditions is shown. The results are applied for the proof of existence of a weak solution of an equation describing stochastic harmonic oscillator. Secondly, the Girsanov-type theorem is used to find the maximum likelihood scalar estimator that appears in the drift of an SDE with additive noise. 1

We approximate the solution of a quasilinear stochastic partial differential equa- tion driven by fractional Brownian motion B_H(t); H in (0,1), which was calculated via fractional White Noise calculus, see [5].

This work is a study of complex many-body systems with non-trivial interactions. Many such systems can be described with models that are much simpler than the real thing but which can still give good insight into the behavior of realistic systems. We take a look at two such systems. The first part looks at a model that elucidates the variety of magnetic phases observed in rare-earth heterostructures at low temperatures: the six-state clock model. We use an ANNNI-like model Hamiltonian that has a three-dimensional parameter space and yields two-dimensional multiphase regions in this space. A low-temperature expansion of the free energy reveals an example of Villain's `order from disorder' [81, 60] when an infinitesimal temperature breaks the ground-state degeneracy. The next part of our work describes biological systems. Using ECIS (Electric Cell-Substrate Impedance Sensing), we are able to extract complex impedance time series from a confluent layer of live cells. We use simple statistics to characterize the behavior of cells in these experiments. We compare experiment with models of fractional Brownian motion and random walks with persistence. We next detect differences in the behavior of single cell types in a toxic environment. Finally, we develop a very simple model of micromotion that helps explain the types of interactions responsible for the long-term and short-term correlations seen in the power spectra and autocorrelation curves extracted from the times series produced from the experiments.

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yes / n this paper, a powerful face recognition system for authentication and identification tasks is presented and a new facial feature extraction approach is proposed. A novel feature extraction method based on combining the characteristics of the Curvelet transform and Fractal dimension transform is proposed. The proposed system consists of four stages. Firstly, a simple preprocessing algorithm based on a sigmoid function is applied to standardize the intensity dynamic range in the input image. Secondly, a face detection stage based on the Viola-Jones algorithm is used for detecting the face region in the input image. After that, the feature extraction stage using a combination of the Digital Curvelet via wrapping transform and a Fractal Dimension transform is implemented. Finally, the K-Nearest Neighbor (K-NN) and Correlation Coefficient (CC) Classifiers are used in the recognition task. Lastly, the performance of the proposed approach has been tested by carrying out a number of experiments on three well-known datasets with high diversity in the facial expressions: SDUMLA-HMT, Faces96 and UMIST datasets. All the experiments conducted indicate the robustness and the effectiveness of the proposed approach for both authentication and identification tasks compared to other established approaches.

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Yes / The most recent face recognition systems are mainly dependent on feature representations obtained using either local handcrafted-descriptors, such as local binary patterns (LBP), or use a deep learning approach, such as deep belief network (DBN). However, the former usually suffers from the wide variations in face images, while the latter usually discards the local facial features, which are proven to be important for face recognition. In this paper, a novel framework based on merging the advantages of the local handcrafted feature descriptors with the DBN is proposed to address the face recognition problem in unconstrained conditions. Firstly, a novel multimodal local feature extraction approach based on merging the advantages of the Curvelet transform with Fractal dimension is proposed and termed the Curvelet–Fractal approach. The main motivation of this approach is that theCurvelet transform, a newanisotropic and multidirectional transform, can efficiently represent themain structure of the face (e.g., edges and curves), while the Fractal dimension is one of the most powerful texture descriptors for face images. Secondly, a novel framework is proposed, termed the multimodal deep face recognition (MDFR)framework, to add feature representations by training aDBNon top of the local feature representations instead of the pixel intensity representations. We demonstrate that representations acquired by the proposed MDFR framework are complementary to those acquired by the Curvelet–Fractal approach. Finally, the performance of the proposed approaches has been evaluated by conducting a number of extensive experiments on four large-scale face datasets: the SDUMLA-HMT, FERET, CAS-PEAL-R1, and LFW databases. The results obtained from the proposed approaches outperform other state-of-the-art of approaches (e.g., LBP, DBN, WPCA) by achieving new state-of-the-art results on all the employed datasets.

Stochastic Integrals Driven by Isonormal Gaussian Processes and Applications Master Thesis - Petr Čoupek Abstract In this thesis, we introduce a stochastic integral of deterministic Hilbert space valued functions driven by a Gaussian process of the Volterra form βt = t 0 K(t, s)dWs, where W is a Brownian motion and K is a square integrable kernel. Such processes generalize the fractional Brownian motion BH of Hurst parameter H ∈ (0, 1). Two sets of conditions on the kernel K are introduced, the singular case and the regular case, and, in particular, the regular case is studied. The main result is that the space H of β-integrable functions can be, in the strictly regular case, embedded in L 2 1+2α ([0, T]; V ) which corresponds to the space L 1 H ([0, T]) for the fractional Brownian mo- tion. Further, the cylindrical Gaussian Volterra process is introduced and a stochastic integral of deterministic operator-valued functions, driven by this process, is defined. These results are used in the theory of stochastic differential equations (SDE), in particular, measurability of a mild solution of a given SDE is proven.