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This paper aims to study fractional Brownian motion and its applications to nonlinear stochastic integral equations. Bernstein polynomials have been applied to obtain the numerical results of the nonlinear fractional stochastic integral equations.

Bernstein polynomials have been used to obtain the numerical solutions of nonlinear fractional stochastic integral equations. The fractional stochastic operational matrix based on Bernstein polynomial has been used to discretize the nonlinear fractional stochastic integral equation. Convergence and error analysis of the proposed method have been discussed.

Two illustrated examples have been presented to justify the efficiency and applicability of the proposed method. The corresponding obtained numerical results have been compared with the exact solutions to establish the accuracy and efficiency of the proposed method.

To the best of the authors’ knowledge, nonlinear stochastic Itô–Volterra integral equation driven by fractional Brownian motion has been for the first time solved by using Bernstein polynomials. The obtained numerical results well establish the accuracy and efficiency of the proposed method.

To define the main elements of a formal calculus which deals with fractional Brownian motion (fBm), and to examine its prospects of applications in systems science.

The approach is based on a generalization of the Maruyama's notation. The key is the new Taylor's series of fractional order f(x+h)=E_α(h^αD^α)f(x), where E_α( · ) is the Mittag‐Leffler function.

As illustrative applications of this formal calculus in systems science, one considers the linear quadratic Gaussian problem with fractal noises, the analysis of the equilibrium position of a system disturbed by a local fractal time, and a model of growing which involves fractal noises. And then, one examines what happens when one applies the maximum entropy principle to systems involving fBms (or shortly fractals).

The framework of this paper is applied mathematics and engineering mathematics, and the results so obtained allow the practical analysis of stochastic dynamics subject to fractional noises.

The direct prospect of application of this approach is the analysis of some stock markets dynamics and some biological systems.

The fractional Taylor's series is new and thus so are all its implications.

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 explain why fractal, self‐similarity, and fractional Brownian motions are so pervasive in human systems.

The analysis involves mainly relative observation, Minkowskian observation, Euclidean observation, and fractional calculus.

It is shown that observation with informational invariance, which is a modeling of subjectivity, creates fractal, and self‐similarity.

This result could have an application to the quantitative analysis of volatility in finance, for instance.

The paper supports the use of fractional dynamics to describe human systems.

The paper provides practical arguments that may explain why fractals are so pervasive in natural science, and mainly in systems involving human factors.

The theory which is addressed in this paper is that we should not be surprised to come across fractals in the analysis of some dynamic systems involving human factors. Moreover, in substance, fractals in human behaviour is acceptable. For the convenience of the reader a primary background to some models of fractional Brownian motions which can be found in the literature is given, and then the main features of the complex‐valued model, via a random walk in the complex plane, recently introduced by the author are recalled. The practical meaning of the model is exhibited. The parallel of the central limit theorem here is Levy’s stability. If it is supposed that human decision‐makers work via an observation process which combines the Heisenberg principle and a quantization principle in the measurement, then fractal dynamics appears to be quite in order. The relation with the theory of relative information is exhibited. The conjecture is then the following: could this model explain why fractals appear in finance, for instance?

This paper seeks to characterize the behavior of the naira/dollar foreign exchange rate series over the period 1999 through 2006 to determine if the process generating the series has long memory which is a special case of fractional Brownian motion. The existence of long memory contradicts the notion of market efficiency.

The paper employs the modified rescaled range R/S test which is proposed by Lo to test the null hypothesis that daily and weekly NGN/USD exchange rates from 1999 through 2006 exhibit short‐memory process. The second test that was also employed is the Geweke‐Porter‐Hubak (GPH) test which was refined by Hurvich et al.

The results show that long memory is present in daily and weekly foreign exchange level series of the Nigerian naira for the period sampled. This evidence implies that the Nigerian foreign exchange market may not be efficient. Thus, it is possible for investors to realize abnormal profit by taking an investment position based on predicted exchange rates. The results reported in this paper are also indicative of a deviation from long‐run PPP.

This paper is the first to empirically apply the modified R/S and GPH tests to explore the existence of long‐memory process in a country study of foreign exchange series using data from Nigeria.

The Hurst exponent has been very important in telling the difference between fractal signals and explaining their significance. For estimators of the Hurst exponent, accuracy and efficiency are two inevitable considerations. The main purpose of this study is to raise the execution efficiency of the existing estimators, especially the fast maximum likelihood estimator (MLE), which has optimal accuracy.

A two-stage procedure combining a quicker method and a more accurate one to estimate the Hurst exponent from a large to small range will be developed. For the best possible accuracy, the data-induction method is currently ideal for the first-stage estimator and the fast MLE is the best candidate for the second-stage estimator.

For signals modeled as discrete-time fractional Gaussian noise, the proposed two-stage estimator can save up to 41.18 per cent the computational time of the fast MLE while remaining almost as accurate as the fast MLE, and even for signals modeled as discrete-time fractional Brownian motion, it can also save about 35.29 per cent except for smaller data sizes.

The proposed two-stage estimation procedure is a novel idea. It can be expected that other fields of parameter estimation can apply the concept of the two-stage estimation procedure to raise computational performance while remaining almost as accurate as the more accurate of two estimators.

The purpose of this paper is to review three papers in this issue and contribute new results on commodity futures prices and volume using wavelet analysis.

The paper uses time series econometrics including variance ratio tests, fractional integration estimators, and wavelet transforms.

The role of time horizon is emphasized in the discussion of the three papers, and wavelet methods are shown to be a useful tool to better understand time horizon-specific risk. Moreover, changes in the time horizon of futures trading are documented and discussed.

In addition to discussing three papers on quantitative finance for agricultural commodities, this paper also looks at how the analysis and management of short-term and long-term risk may differ. To this end, wavelet transform-based time series methods are reviewed and applied.

Turvey (2007, Physica A) introduced a scaled variance ratio procedure for testing the random walk hypothesis (RWH) for financial time series by estimating Hurst coefficients for a fractional Brownian motion model of asset prices. The purpose of this paper is to extend his work by making the estimation procedure robust to heteroskedasticity and by addressing the multiple hypothesis testing problem.

Unbiased, heteroskedasticity consistent, variance ratio estimates are calculated for end of day price data for eight time lags over 12 agricultural commodity futures (front month) and 40 US equities from 2000-2014. A bootstrapped stepdown procedure is used to obtain appropriate statistical confidence for the multiplicity of hypothesis tests. The variance ratio approach is compared against regression-based testing for fractionality.

Failing to account for bias, heteroskedasticity, and multiplicity of testing can lead to large numbers of erroneous rejections of the null hypothesis of efficient markets following an independent random walk. Even with these adjustments, a few futures contracts significantly violate independence for short lags at the 99 percent level, and a number of equities/lags violate independence at the 95 percent level. When testing at the asset level, futures prices are found not to contain fractional properties, while some equities do.

Only a subsample of futures and equities, and only a limited number of lags, are evaluated. It is possible that multiplicity adjustments for larger numbers of tests would result in fewer rejections of independence.

This paper provides empirical evidence that violations of the RWH for financial time series are likely to exist, but are perhaps less common than previously thought.

Empirical findings on interest rate dynamics imply that short rates show some long memories and non-Markovian. It is well-known that fractional Brownian motion (IBm) is a proper candidate for modelling this empirical phenomena. IBm. however. is not a semimartingale process. For this reason. it is very hard to apply such processes for asset price modelling.

Without using Ito formula, we investigate the IBm interest rate theory‘ We obtain a pure discount bond price. and Greeks by using Malllavin calculus.