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The discrete Fourier transform (dft) of a fractional process is studied. An exact representation of the dft is given in terms of the component data, leading to the frequency domain form of the model for a fractional process. This representation is particularly useful in analyzing the asymptotic behavior of the dft and periodogram in the nonstationary case when the memory parameter d≥12. Various asymptotic approximations are established including some new hypergeometric function representations that are of independent interest. It is shown that smoothed periodogram spectral estimates remain consistent for frequencies away from the origin in the nonstationary case provided the memory parameter d < 1. When d = 1, the spectral estimates are inconsistent and converge weakly to random variates. Applications of the theory to log periodogram regression and local Whittle estimation of the memory parameter are discussed and some modified versions of these procedures are suggested for nonstationary cases.

This chapter derives asymptotic properties of the least squares (LS) estimator of the autoregressive (AR) parameter in local to unity processes with errors being fractional Gaussian noise (FGN) with the Hurst parameter H∈(0,1). It is shown that the estimator is consistent for all values of H∈(0,1). Moreover, the rate of convergence is n−1 when H∈[0.5,1). The rate of convergence is n−2H when H∈(0,0.5). Furthermore, the limiting distribution of the centered LS estimator depends on H. When H=0.5, the limiting distribution is the same as that obtained in Phillips (1987a) for the local to unity model with errors for which the standard functional central limit theorem is applicable. When H > 0.5 or when H < 0.5, the limiting distributions are new to the literature. The asymptotic properties of the LS estimator with fitted intercept are also derived. Simulation studies are performed to check the reliability of the asymptotic approximation for different values of sample size.

Joon Y. Park is one of the pioneers in developing nonlinear cointegrating regression. Since his initial work with Phillips (Park & Phillips, 2001) in the area, the past two decades have witnessed a surge of interest in modeling nonlinear nonstationarity in macroeconomic and financial time series, including parametric, nonparametric and semiparametric specifications of such models. These developments have provided a framework of econometric estimation and inference for a wide class of nonlinear, nonstationary relationships. In honor of Joon Y. Park, this chapter contributes to this area by exploring nonparametric estimation of functional-coefficient cointegrating regression models where the structural equation errors are serially dependent and the regressor is endogenous. The self-normalized local kernel and local linear estimators are shown to be asymptotic normal and to be pivotal upon an estimation of co-variances. Our new results improve those of Cai et al. (2009) and open up inference by conventional nonparametric method to a wide class of potentially nonlinear cointegrated relations.

The purpose of this paper is to propose a novel nonlocal fractal calculus scheme dedicated to the analysis of fractal electrical circuit, namely, the generalized nonlocal fractal calculus.

For being generalized, an arbitrary kernel function has been adopted. The condition on order has been derived so that it is not related to the γ-dimension of the fractal set. The fractal Laplace transforms of our operators have been derived.

Unlike the traditional power law kernel-based nonlocal fractal calculus operators, ours are generalized, consistent with the local fractal derivative and use higher degree of freedom. As intended, the proposed nonlocal fractal calculus is applicable to any kind of fractal electrical circuit. Thus, it has been found to be a more efficient tool for the fractal electrical circuit analysis than any previous fractal set dedicated calculus scheme.

A fractal calculus scheme that is more efficient for the fractal electrical circuit analysis than any previous ones has been proposed in this work.

The purpose of this study is to analyse and compile the literature on various option pricing models (OPM) or methodologies. The report highlights the gaps in the existing literature review and builds recommendations for potential scholars interested in the subject area.

In this study, the researchers used a systematic literature review procedure to collect data from Scopus. Bibliometric and structured network analyses were used to examine the bibliometric properties of 864 research documents.

As per the findings of the study, publication in the field has been increasing at a rate of 6% on average. This study also includes a list of the most influential and productive researchers, frequently used keywords and primary publications in this subject area. In particular, Thematic map and Sankey’s diagram for conceptual structure and for intellectual structure co-citation analysis and bibliographic coupling were used.

Based on the conclusion presented in this paper, there are several potential implications for research, practice and society.

This study provides useful insights for future research in the area of OPM in financial derivatives. Researchers can focus on impactful authors, significant work and productive countries and identify potential collaborators. The study also highlights the commonly used OPMs and emerging themes like machine learning and deep neural network models, which can inform practitioners about new developments in the field and guide the development of new models to address existing limitations.

The accurate pricing of financial derivatives has significant implications for society, as it can impact the stability of financial markets and the wider economy. The findings of this study, which identify the most commonly used OPMs and emerging themes, can help improve the accuracy of pricing and risk management in the financial derivatives sector, which can ultimately benefit society as a whole.

It is possibly the initial effort to consolidate the literature on calibration on option price by evaluating and analysing alternative OPM applied by researchers to guide future research in the right direction.

The fractional order HIV model has an important role in biological science. To study the HIV model in a better way, the model is presented with the help of Atangana- Baleanu operator which is in Caputo sense. Also, the characteristics of the solutions are described briefly with the help of the advance numerical techniques for the different values of fractional order derivatives. This paper aims to discuss the aforementioned objectives.

In this work, Adams-Bashforth method and Euler method are used to get the solution of the HIV model. These are the important numerical methods. The comparison results also are described with the physical meaning of the solutions of the model.

HIV model is analyzed under the view of fractional and AB derivative in Atangana-Baleanu-Caputo sense. The uniqueness of the solution is proved by using Banach Fixed point. The solution is derived with the help of Sumudu transform. Further, the authors employed fractional Adam-Bashforth method and Euler method to enumerate numerical results. The authors have used several values of fractional orders to present the outcomes graphically. The above calculations have been done with the help of MATLAB (R2016a). The numerical scheme used in the proposed study is valid and fruitful, and the same can be used to explore other real issues.

This investigation can be done for the real data sets.

This paper aims to express the solution of the HIV model in a better way with the effect of non-locality, this work is very useful.

In this work, HIV model is developed with the help of Atangana- Baleanu operator in Caputo sense. By using Banach Fixed point, the authors proved that the solution is unique. Also, the solution is presented with the help of Sumudu transform. The behaviors of the solutions are checked for different values of fractional order derivatives with the physical meaning with help of the Adam-Bashforth method and the Euler method.

To propose a new investment-income valuation model by real options approach (ROA) for old community renewal (OCR) projects, which could help the government attract private capital's participation.

The new model is proposed by identifying the types of options private capital has in the OCR project, selecting the option model most suitable for private capital investment decisions, improving the valuation model through the triangular fuzzy numbers to take into account the uncertainty and flexibility, and demonstrating the feasibility of the calculation model through an actual OCR project case.

The new model can valuate OCR projects more accurately based on considering uncertainty and flexibility, compared with conventional methods that often underestimate the value of OCR projects.

The investment-income of OCR projects shall be re-valuated from the lens of real options, which could help reveal more real benefits beyond the capital growth of OCR projects, enable the government to attract private capital's investment in OCR, and alleviate government fiscal pressure.

The proposed OCR-oriented investment-income valuation model systematically analyzes the applicability of real option value (ROV) to OCR projects, innovatively integrates the ROV and the net present value (NPV) as expanded net present value (ENPV), and accurately evaluate real benefits in comparison with existing models. Furthermore, the newly proposed model holds the potential to be transferred to various social welfare projects as a tool to attract private capital's participation.

The purpose of this research is to analyze the Bitcoin (BTC) and Ether (ETH) long memory and conditional volatility.

The empirical approach includes ARFIMA-HYGARCH and ARFIMA-FIGARCH, both models under Student‘s t-distribution, during the period (ETH: November 9, 2017 to November 25, 2021 and BTC: September 17, 2014 to November 25, 2021).

Findings suggest that ARFIMA-HYGARCH is the best model to analyze BTC volatility, and ARFIMA-FIGARCH is the best approach to model ETH volatility. Empirical evidence also confirms the existence of long memory on returns and on BTC volatility parameters. Results evidence that the models proposed are not as suitable for modeling ETH volatility as they are for the BTC.

Findings allow to confirm the fractal market hypothesis in BTC market. The data confirm that, despite the impact of the Covid-19 crisis, the dynamics of BTC returns, and volatility maintained their patterns, i.e. the way in which they evolve, in relation to the prepandemic era, did not change, but it is rather reaffirmed. Yet, ETH conditional volatility was more affected, as it is apparently higher during Covid-19. The originality of the research lies in the focus of the analysis, the proposed methodology and the variables and periods of study.

The purpose of this paper is to perform Shariah review of Brownian motion that is used for prediction of Islamic stock prices and their volatility.

It uses the Shariah compliant development model guidelines to review the Brownian motion and its applications.

The model of Brownian motion does not involve any variable that renders it non-Shariah compliant; neither all applications of Brownian motion are Shariah compliant. Because the model is based on stochastic properties that involve randomness, therefore the issue of gharar takes the utmost important to handle in the applications of the model. The results need to be analyzed strictly in accordance with the Shariah whether they create any element of gharar or uncertainty in case of expected price and volatility estimates.

The research suffers from the limitation that it analyses only one model of physics, i.e. Brownian motion model from Shariah perspective.

The research opens an area for Shariah analysis of results generated from the application of advanced models of physics on matters related to Islamic financial markets.

The originality of this study stems from the fact that to the best of the authors’ knowledge, it is the first study that extends Shariah guidelines into Financial physics for making the foundations of Islamic econophysics.

The need for precise synthesis of customized designs has resulted in the development of advanced coating processes for modern nanomaterials. Achieving accuracy in these processes requires a deep understanding of thermophysical behavior, rheology and complex chemical reactions. The manufacturing flow processes for these coatings are intricate and involve heat and mass transfer phenomena. Magnetic nanoparticles are being used to create intelligent coatings that can be externally manipulated, making them highly desirable. In this study, a Keller box calculation is used to investigate the flow of a coating nanofluid containing a viscoelastic polymer over a circular cylinder.

The rheology of the coating polymer nanofluid is described using the viscoelastic model, while the effects of nanoscale are accounted for by using Buongiorno’s two-component model. The nonlinear PDEs are transformed into dimensionless PDEs via a nonsimilar transformation. The dimensionless PDEs are then solved using the Keller box method.

The transport phenomena are analyzed through a comprehensive parametric study that investigates the effects of various emerging parameters, including thermal radiation, Biot number, Eckert number, Brownian motion, magnetic field and thermophoresis. The results of the numerical analysis, such as the physical variables and flow field, are presented graphically. The momentum boundary layer thickness of the viscoelastic polymer nanofluid decreases as fluid parameter increases. An increase in mixed convection parameter leads to a rise in the Nusselt number. The enhancement of the Brinkman number and Biot number results in an increase in the total entropy generation of the viscoelastic polymer nanofluid.

Intelligent materials rely heavily on the critical characteristic of viscoelasticity, which displays both viscous and elastic effects. Viscoelastic models provide a comprehensive framework for capturing a range of polymeric characteristics, such as stress relaxation, retardation, stretching and molecular reorientation. Consequently, they are a valuable tool in smart coating technologies, as well as in various applications like supercapacitor electrodes, solar collector receivers and power generation. This study has practical applications in the field of coating engineering components that use smart magnetic nanofluids. The results of this research can be used to analyze the dimensions of velocity profiles, heat and mass transfer, which are important factors in coating engineering. The study is a valuable contribution to the literature because it takes into account Joule heating, nonlinear convection and viscous dissipation effects, which have a significant impact on the thermofluid transport characteristics of the coating.

The momentum boundary layer thickness of the viscoelastic polymer nanofluid decreases as the fluid parameter increases. An increase in the mixed convection parameter leads to a rise in the Nusselt number. The enhancement of the Brinkman number and Biot number results in an increase in the total entropy generation of the viscoelastic polymer nanofluid. Increasing the strength of the magnetic field promotes an increase in the density of the streamlines. An increase in the mixed convection parameter results in a decrease in the isotherms and isoconcentration.