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This paper derives the analytic solutions of the pure discount bond price under the various types of -stable Levy process. It is well-known that only a few cases in-stable Levy process have the moment generating function. This paper extends the model to damped-stable Levy processes, which have artificial stable process with the moment generating function. This paper also extends models to stochastic volatility by time change method of Levy process.

In Islamic finance (IF), the safety-first rule of investing (hifdh al mal) is held to be of utmost importance. In view of the instability in the global financial markets, the IF portfolio manager (mudharib) is committed, according to Sharia, to make use of advanced models and reliable tools. This paper seeks to address these issues.

In this paper, the limitations of the standard models used in the IF industry are reviewed. Then, a framework was set forth for a reliable modeling of the IF markets, especially in extreme events and highly volatile periods. Based on the empirical evidence, the framework offers an improved tool to ameliorate the evaluation of Islamic stock market risk exposure and to reduce the costs of Islamic risk management.

Based on the empirical evidence, the framework offers an improved tool to ameliorate the evaluation of Islamic stock market risk exposure and to reduce the costs of Islamic risk management.

In IF, the portfolio manager – mudharib – according to Sharia, should ensure the adequacy of the mathematical and statistical tools used to model and control portfolio risk. This task became more complicated because of the increase in risk, as measured via market volatility, during the financial crisis that began in the summer of 2007. Sharia condemns the portfolio manager who demonstrates negligence and may hold him accountable for losses for failing to select the proper analytical tools. As Sharia guidelines hold the safety-first principle of investing rule (hifdh al mal) to be of utmost importance, the portfolio manager should avoid speculative investments and strategies that would lead to significant losses during periods of high market volatility.

The purpose of this paper is to argue that a stationary-differenced autoregressive (AR) process with lag greater than 1, AR(q > 1), has certain properties that are consistent with a fractional Brownian motion (fBm). What the authors are interested in is the investigation of approaches to identifying the existence of persistent memory of one form or another for the purposes of simulating commodity (and other asset) prices. The authors show in theory, and with application to agricultural commodity prices the relationship between AR(q) and quasi-fBm.

In this paper the authors develop mathematical relationships in support of using AR(q > 1) processes for simulating quasi-fBm.

From theory the authors show that any AR(q) process is a stationary, self-similar process, with a lag structure that captures the essential elements of scaling and a fractional power law. The authors illustrate through various means the approach, and apply the quasi-fractional AR(q) process to agricultural commodity prices.

While the results can be applied to most time series of commodity prices, the authors limit the evaluation to the Gaussian case. Thus the approach does not apply to infinite-variance models.

The approach to using the structure of an AR(q > 1) model to simulate quasi-fBm is a simple approach that can be applied with ease using conventional Monte Carlo methods.

The authors believe that the approach to simulating quasi-fBm using standard AR(q > 1) models is original. The approach is intuitive and can be applied easily.

The purpose of this paper is to develop an alternative numerical approach for describing fractional diffusion in Cartesian and non‐Cartesian domains using a Monte Carlo random walk scheme. The resulting domain shifting scheme provides a numerical solution for multi‐dimensional steady state, source free diffusion problems with fluxes expressed in terms of Caputo fractional derivatives. This class of problems takes account of non‐locality in transport, expressed through parameters representing both the extent and direction of the non‐locality.

The method described here follows a similar approach to random walk methods previously developed for normal (local) diffusion. The key differences from standard methods are: first, the random shifting of the domain about the point of interest with, second, shift steps selected from non‐symmetric, power‐law tailed, Lévy probability distribution functions.

The domain shifting scheme is verified by comparing predictive solutions to known one‐dimensional and two‐dimensional analytical solutions for fractional diffusion problems. The scheme is also applied to a problem of fractional diffusion in a non‐Cartesian annulus domain. In contrast to the axisymmetric, steady state solution for normal diffusion, a non‐axisymmetric solution results.

This is the first random walk scheme to utilize the concept of allowing the domain to undergo the random walk about a point of interest. Domain shifting scheme solutions of fractional diffusion in non‐Cartesian domains provide an invaluable tool to direct the development of more sophisticated grid based finite element inspired fractional diffusion schemes.

The variance targeting estimator (VTE) for generalized autoregressive conditionally heteroskedastic (GARCH) processes has been proposed as a computationally simpler and misspecification-robust alternative to the quasi-maximum likelihood estimator (QMLE). In this paper we investigate the asymptotic behavior of the VTE when the stationary distribution of the GARCH process has infinite fourth moment. Existing studies of historical asset returns indicate that this may be a case of empirical relevance. Under suitable technical conditions, we establish a stable limit theory for the VTE, with the rate of convergence determined by the tails of the stationary distribution. This rate is slower than that achieved by the QMLE. The limit distribution of the VTE is nondegenerate but singular. We investigate the use of subsampling techniques for inference, but find that finite sample performance is poor in empirically relevant scenarios.

The main purpose of this paper is to estimate the resistance and inductor in the RL electrical circuit when these are unavailable or missing data that it is a concern in electrical engineering. The input voltage is assumed to be corrupted by the noise and the current is observed at discrete time points.

The authors propose a computationally efficient framework for parameters estimation using least square estimator and Bayesian Monte Carlo scheme.

The explicit formulas for least square estimator are derived and the strong consistency of resistance estimator is verified when inductor is a known parameter, then Bayesian estimation of parameters governed by using Markov chain Monte Carlo methods. The applicability of the results is demonstrated by using numerical examples. Several numerical results and figures are presented via Matlab and R programming to illustrate the performance of the estimators.

The paper can be used in various types of electrical engineering real time projects. The projects include electrical circuits, electrical machines theory and drives, especially when the parameters are uncertain that it is a worry in electrical engineering.

To the author's best knowledge, least square and Bayesian estimation of resistance and inductor have not been studied before. The proposed model is nonlinear with respect to inductor (L); therefore the present work has fundamental difference in comparison with the similar models.

This study aims to introduce a methodology for optimal allocation of spinning reserves taking into account load, wind and solar generation by application of the univariate and bivariate parametric models, conventional intra and inter-zonal spinning reserve capacity as well as demand response through utilization of capacity outage probability tables and the equivalent assisting unit approach.

The method uses a novel approach to model wind power generation using the bivariate Farlie–Gumbel–Morgenstern probability density function (PDF). The study also uses the Bayesian network (BN) algorithm to perform the adjustment of spinning reserve allocation, based on the actual unit commitment of the previous hours.

The results show that the utilization of bivariate wind prediction model along with reserve allocation adjustment algorithm improve reliability of the power grid by 2.66% and reduce the total system operating costs by 1.12%.

The method uses a novel approach to model wind power generation using the bivariate Farlie–Gumbel–Morgenstern PDF. The study also uses the BN algorithm to perform the adjustment of spinning reserve allocation, based on the actual unit commitment of the previous hours.

The purpose of this paper is to investigate the effects of irregular unsettling on the smoking model in form of the stochastic model as in the deterministic model these effects are neglected for simplicity.

In this research, the authors investigate a stochastic smoking system in which the contact rate is perturbed by Lévy noise to control the trend of smoking. First, present the formulation of the stochastic model and study the dynamics of the deterministic model. Then the global positive solution of the stochastic system is discussed. Further, extinction and the persistence of the proposed system are presented on the base of the reproductive number.

The authors discuss the dynamics of the deterministic smoking model form and further present the existence and uniqueness of non-negative global solutions for the stochastic system. Some previous study’s mentioned in the Introduction can be improved with the help of obtaining results, graphically present in this manuscript. In this regard, the authors present the sufficient conditions for the extinction of smoking for reproductive number is less than 1.

In this work, the authors investigated the dynamic stochastic smoking model with non-Gaussian noise. The authors discussed the dynamics of the deterministic smoking model form and further showed for the stochastic system the existence and uniqueness of the non-negative global solution. Some previous study’s mentioned in the Introduction can be improved with the help of obtained results, clearly shown graphically in this manuscript. In this regard, the authors presented the sufficient conditions for the extinction of smoking, if <1, which can help in the control of smoking. Motivated from this research soon, the authors will extent the results to propose new mathematical models for the smoking epidemic in the form of fractional stochastic modeling. Especially, will investigate the effective strategies for control smoking throughout the world.

This study is helpful in the control of smoking throughout the world.

Financial asset return series usually exhibit nonnormal characteristics such as high peaks, heavy tails and asymmetry. Traditional risk measures like standard deviation or variance are inadequate for nonnormal distributions. Value at Risk (VaR) is consistent with people's psychological perception of risk. The asymmetric Laplace distribution (ALD) captures the heavy-tailed and biased features of the distribution. VaR is therefore used as a risk measure to explore the problem of VaR-based asset pricing. Assuming returns obey ALD, the study explores the impact of high peaks, heavy tails and asymmetric features of financial asset return data on asset pricing.

A VaR-based capital asset pricing model (CAPM) was constructed under the ALD that follows the logic of the classical CAPM and derive the corresponding VaR-β coefficients under ALD.

ALD-based VaR exhibits a minor tail risk than VaR under normal distribution as the mean increases. The theoretical derivation yields a more complex capital asset pricing formula involving β coefficients compared to the traditional CAPM.The empirical analysis shows that the CAPM under ALD can reflect the β-return relationship, and the results are robust. Finally, comparing the two CAPMs reveals that the β coefficients derived in this paper are smaller than those in the traditional CAPM in 69–80% of cases.

The paper uses VaR as a risk measure for financial time series data following ALD to explore asset pricing problems. The findings complement existing literature on the effects of high peaks, heavy tails and asymmetry on asset pricing, providing valuable insights for investors, policymakers and regulators.