Option Pricing under the Subordinated Market Models

This paper aims to study option pricing problem under the subordinated Brownian motion. Firstly, we prove that the subordinated Brownian motion controlled by the fractional diffusion equation has many financial properties, such as self-similarity, leptokurtic, and long memory, which indicate that the fractional calculus can describe the financial data well. Then, we investigate the option pricing under the assumption that the stock price is driven by the subordinated Brownian motion. The closed-form pricing formula for European options is derived. In the comparison with the classic Black–Sholes model, we find the option prices become higher, and the “volatility smiles” phenomenon happens in the proposed model. Finally, an empirical analysis is performed to show the validity of these results.

Penerapan Modifikasi Fraktal Segitiga Sierpinski pada Ragam Hias Geometris Tumpal

Sierpinski’s triangular fractal is a linear fractal that has self-similarity, which is identical until infinite iterations. This research aims to develop the Tumpal geometric ornaments with the implementation of modified Sierpinski’s triangular fractal. There are three algorithms that will be used. First, an algorithm to modify the Sierpinski triangle. The isosceles triangle is divided into nine congruent triangles. Then randomly selected several triangles to be left blank. Do the same way to the triangle that still exists until some iteration. Second, modeling the base frames. Third, fill the basic frame from the second algorithm with the modified Sierpinski's triangular fractal from the first algorithm into a motif. The results are various Tumpal geometric motifs with the implementation of modified Sierpinski’s triangular fractal. Keywords: linear fractal, Sierpinski’s triangular fractal, ornament, Tumpal geometric Abstrak Fraktal segitiga Sierpinski merupakan fraktal linier yang memiliki sifat self-similarity, yaitu identik sampai pada iterasi tak terhingga. Penelitian ini bertujuan untuk mengembangkan ragam hias geometris Tumpal dengan penerapan modifikasi fraktal segitiga Sierpinski. Ada tiga algoritma yang akan digunakan. Pertama, algoritma yang bertujuan untuk memodifikasi segitiga Sierpinski. Data awal berupa segitiga samakaki yang dibagi menjadi sembilan segitiga kongruen. Kemudian dipilih secara acak beberapa segitiga yang akan dikosongkan. Pada segitiga yang masih berisi dilakukan hal yang sama Kedua, modelisasi bingkai dasar. Ketiga, mengisi bingkai dasar hasil algoritma kedua dengan modifikasi segitiga Sierpinski hasil algoritma pertama sehingga menjadi sebuah motif. Hasil penelitian yang diperoleh adalah beragam motif geometris Tumpal dengan penerapan modifikasi segitiga Sierpinski. Kata Kunci: fraktal linier, segitiga Sierpinski, ragam hias, geometris Tumpal

Fractal-Based Analysis of fMRI BOLD Signal During Naturalistic Viewing Conditions

Background: Temporal fractals are characterized by prominent scale-invariance and self-similarity across time scales. Monofractal analysis quantifies this scaling behavior in a single parameter, the Hurst exponent (H). Higher H reflects greater correlation in the signal structure, which is taken as being more fractal. Previous fMRI studies have observed lower H during conventional tasks relative to resting state conditions, and shown that H is negatively correlated with task difficulty and novelty. To date, no study has investigated the fractal dynamics of BOLD signal during naturalistic conditions.Methods: We performed fractal analysis on Human Connectome Project 7T fMRI data (n = 72, 41 females, mean age 29.46 ± 3.76 years) to compare H across movie-watching and rest.Results: In contrast to previous work using conventional tasks, we found higher H values for movie relative to rest (mean difference = 0.014; p = 5.279 × 10−7; 95% CI [0.009, 0.019]). H was significantly higher in movie than rest in the visual, somatomotor and dorsal attention networks, but was significantly lower during movie in the frontoparietal and default networks. We found no cross-condition differences in test-retest reliability of H. Finally, we found that H of movie-derived stimulus properties (e.g., luminance changes) were fractal whereas H of head motion estimates were non-fractal.Conclusions: Overall, our findings suggest that movie-watching induces fractal signal dynamics. In line with recent work characterizing connectivity-based brain state dynamics during movie-watching, we speculate that these fractal dynamics reflect the configuring and reconfiguring of brain states that occurs during naturalistic processing, and are markedly different than dynamics observed during conventional tasks.

Region-specific complexity of the intracranial EEG in the sleeping human brain

As the brain is a complex system with occurrence of self-similarity at different levels, a dedicated analysis of the complexity of brain signals is of interest to elucidate the functional role of various brain regions across the various stages of vigilance. We exploited intracranial electroencephalogram data from 38 cortical regions using the Higuchi fractal dimension (HFD) as measure to assess brain complexity, on a dataset of 1772 electrode locations. HFD values depended on sleep stage and topography. HFD increased with higher levels of vigilance, being highest during wakefulness in the frontal lobe. HFD did not change from wake to stage N2 in temporo-occipital regions. The transverse temporal gyrus was the only area in which the HFD did not differ between any two vigilance stages. Interestingly, HFD of wakefulness and stage R were different mainly in the precentral gyrus, possibly reflecting motor inhibition in stage R. The fusiform and parahippocampal gyri were the only areas showing no difference between wakefulness and N2. Stages R and N2 were similar only for the postcentral gyrus. Topographical analysis of brain complexity revealed that sleep stages are clearly differentiated in fronto-central brain regions, but that temporo-occipital regions sleep differently.

Numerical simulation of reynolds number effect of a free incompressible isothermal turbulent coaxial jet

This paper studies the effect of Reynolds number on a two-dimensional free incompressible isothermal coaxial turbulent jet over a range of high Reynolds numbers. This is necessary because of its application in noise control and mixing. The Reynolds numbers at the nozzle exit were 9824, 19648, 29472, 39296 and 49120. The models were designed in ANSYS Design Modeler and the numerical simulation was done using a finite volume based Computational Fluid Dynamics (CFD) in ANSYS FLUENT using the two-dimensional Realizable turbulence model. The Governing equations were discretized using the finite volume method with the solution based on the PISO algorithm. The decay of centerline velocity, turbulent kinetic energy profile, the radial profile of axial velocity and similarity profile were investigated along the flow direction. Contour plot indicates that the velocity is high at the jet exit and decreases downstream due to the rapid mixing of the inner and outer jet and the surrounding fluid. It is found generally that Reynolds number plays significant role especially before self-similarity region. The result shows that increasing the Reynolds number give rise to more turbulence which in turn decreases the potential core length, turbulent kinetic energy and enhances the mixing of the fluid. However, at the jet exit, the flow with the lowest Reynolds number has the highest turbulent kinetic energy because it suffers the greater shear. The spreading of the jet was more or less independent of the Reynolds number beyond the self-similarity region. It is also found that the velocity profile is brought to congruence at about z/D=25 for the Reynolds numbers considered

Fast Terahertz Imaging Model Based on Group Sparsity and Nonlocal Self-Similarity

In order to solve the problems of long-term image acquisition time and massive data processing in a terahertz time domain spectroscopy imaging system, a novel fast terahertz imaging model, combined with group sparsity and nonlocal self-similarity (GSNS), is proposed in this paper. In GSNS, the structure similarity and sparsity of image patches in both two-dimensional and three-dimensional space are utilized to obtain high-quality terahertz images. It has the advantages of detail clarity and edge preservation. Furthermore, to overcome the high computational costs of matrix inversion in traditional split Bregman iteration, an acceleration scheme based on conjugate gradient method is proposed to solve the terahertz imaging model more efficiently. Experiments results demonstrate that the proposed approach can lead to better terahertz image reconstruction performance at low sampling rates.

Fractal Dimension Analysis to Detect the Progress of Cancer Using Transmission Optical Microscopy

Fractal dimension, a measure of self-similarity in a structure, is a powerful physical parameter for the characterization of structural property of many partially filled disordered materials. Biological tissues are fractal in nature and reports show a change in self-similarity associated with the progress of cancer, resulting in changes in their fractal dimensions. Here, we report that fractal dimension measurement is a potential technique for the detection of different stages of cancer using transmission optical microscopy. Transmission optical microscopy of a thin tissue sample produces intensity distribution patterns proportional to its refractive index pattern, representing its mass density distribution. We measure fractal dimension detection of different cancer stages and find its universal feature. Many deadly cancers are difficult to detect in their early to different stages due to the hard-to-reach location of the organ and/or lack of symptoms until very late stages. To study these deadly cancers, tissue microarray (TMA) samples containing different stages of cancers are analyzed for pancreatic, breast, colon, and prostate cancers. The fractal dimension method correctly differentiates cancer stages in progressive cancer, raising possibilities for a physics-based accurate diagnosis method for cancer detection.

Experimental study on radial gravity currents flowing in a vegetated channel

We present an experimental study of gravity currents in a cylindrical geometry, in the presence of vegetation. Forty tests were performed with a brine advancing in a fresh water ambient fluid, in lock release, and with a constant and time-varying flow rate. The tank is a circular sector of angle $30^\circ$ with radius equal to 180 cm. Two different densities of the vegetation were simulated by vertical plastic rods with diameter $D=1.6\ \textrm{cm}$ . We marked the height of the current as a function of radius and time and the position of the front as a function of time. The results indicate a self-similar structure, with lateral profiles that after an initial adjustment collapse to a single curve in scaled variables. The propagation of the front is well described by a power law function of time. The existence of self-similarity on an experimental basis corroborates a simple theoretical model with the following assumptions: (i) the dominant balance is between buoyancy and drag, parameterized by a power law of the current velocity $\sim |u|^{\lambda-1}u$ ; (ii) the current advances in shallow-water conditions; and (iii) ambient-fluid dynamics is negligible. In order to evaluate the value of ${\lambda}$ (the only tuning parameter of the theoretical model), we performed two additional series of measurements. We found that $\lambda$ increased from 1 to 2 while the Reynolds number increased from 100 to approximately $6\times10^3$ , and the drag coefficient and the transition from $\lambda=1$ to $\lambda=2$ are quantitatively affected by D, but the structure of the model is not.

On the self-similarity index of 𝑝-adic analytic pro-𝑝 groups

Let 𝑝 be a prime. We say that a pro-𝑝 group is self-similar of index p k p^{k} if it admits a faithful self-similar action on a p k p^{k} -ary regular rooted tree such that the action is transitive on the first level. The self-similarity index of a self-similar pro-𝑝 group 𝐺 is defined to be the least power of 𝑝, say p k p^{k} , such that 𝐺 is self-similar of index p k p^{k} . We show that, for every prime p ⩾ 3 p\geqslant 3 and all integers 𝑑, there exist infinitely many pairwise non-isomorphic self-similar 3-dimensional hereditarily just-infinite uniform pro-𝑝 groups of self-similarity index greater than 𝑑. This implies that, in general, for self-similar 𝑝-adic analytic pro-𝑝 groups, one cannot bound the self-similarity index by a function that depends only on the dimension of the group.