On the Integral of the Fractional Brownian Motion and Some Pseudo-Fractional Gaussian Processes

Abundo, Mario; Pirozzi, Enrica

doi:10.3390/math7100991

Cited by 11 publications

(4 citation statements)

References 26 publications

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“…In this article, the FGN model is chosen to simulate fractal processes. This model is fully described by two parameters only, namely by a variance and Hurst exponent [14,[28][29][30][31].…”

Section: Related Workmentioning

confidence: 99%

Methods for Mathematical Analysis of Simulated and Real Fractal Processes with Application in Cardiology

et al. 2022

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In the article, a comparative analysis is performed regarding the accuracy parameter in determining the degree of self-similarity of fractal processes between the following methods: Variance-Time plot, Rescaled Range (R/S), Wavelet-based, Detrended Fluctuation Analysis (DFA) and Multifractal Detrended Fluctuation Analysis (MFDFA). To evaluate the methods, fractal processes based of Fractional Gaussian Noise were simulated and the dependence between the length of the simulated process and the degree of self-similarity was investigated by calculating the Hurst exponent (H > 0.5). It was found that the Wavelet-based, DFA and MFDFA methods, with a process length greater than 214 points, have a relative error of the Hurst exponent is less than 1%. A methodology for the Wavelet-based method related to determining the size of the scale and the wavelet algorithm was proposed, and it was investigated in terms of the exact determination of the Hurst exponent of two algorithms: Haar and Daubechies with different number of coefficients and different values of the scale. Based on the analysis, it was determined that the Daubechies algorithm with 10 coefficients and scale (i = 2, j = 10) has a relative error of less than 0.5%. The three most accurate methods are applied to the study of real cardiac signals of two groups of people: healthy and unhealthy (arrhythmia) subjects. The results of the statistical analysis, using the t-test, show that the proposed methods can distinguish the two studied groups and can be used for diagnostic purposes.

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“…In this article, the FGN model is chosen to simulate fractal processes. This model is fully described by two parameters only, namely by a variance and Hurst exponent [14,[28][29][30][31].…”

Section: Related Workmentioning

confidence: 99%

Methods for Mathematical Analysis of Simulated and Real Fractal Processes with Application in Cardiology

et al. 2022

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“…Geometric fractional Brownian motion has been studied extensively, with many studies identifying potential applications in the context of neuronal models [58]. It was validated by a rigorous statistical test with added white Gaussian noise based on the autocovariance function [59] and has been applied in studies both for stock indexes [60] and for the price of financial securities [61], in the case of cryptocurrencies [62], in the case of modeling epidemic diseases, such as coronavirus [63], and for the price of goods [50].…”

Section: Introductionmentioning

confidence: 99%

Efficient or Fractal Market Hypothesis? A Stock Indexes Modelling Using Geometric Brownian Motion and Geometric Fractional Brownian Motion

et al. 2021

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In this article, we propose a test of the dynamics of stock market indexes typical of the US and EU capital markets in order to determine which of the two fundamental hypotheses, efficient market hypothesis (EMH) or fractal market hypothesis (FMH), best describes market behavior. The article’s major goal is to show how to appropriately model return distributions for financial market indexes, specifically which geometric Brownian motion (GBM) and geometric fractional Brownian motion (GFBM) dynamic equations best define the evolution of the S&P 500 and Stoxx Europe 600 stock indexes. Daily stock index data were acquired from the Thomson Reuters Eikon database during a ten-year period, from January 2011 to December 2020. The main contribution of this work is determining whether these markets are efficient (as defined by the EMH), in which case the appropriate stock indexes dynamic equation is the GBM, or fractal (as described by the FMH), in which case the appropriate stock indexes dynamic equation is the GFBM. In this paper, we consider two methods for calculating the Hurst exponent: the rescaled range method (RS) and the periodogram method (PE). To determine which of the dynamics (GBM, GFBM) is more appropriate, we employed the mean absolute percentage error (MAPE) method. The simulation results demonstrate that the GFBM is better suited for forecasting stock market indexes than the GBM when the analyzed markets display fractality. However, while these findings cannot be generalized, they are verisimilar.

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“…Traditionally, classical Brownian motion (BM) is a fundamental theory for monitoring outcome effects in clinical trials, including those with CAR designs [10][11][12][13][14]. It has been proved that the sequential test statistics of covariate adaptive clinical trials follow Brownian motion asymptotically under some regularized conditions [15].…”

Section: Introductionmentioning

confidence: 99%

Estimating Conditional Power for Sequential Monitoring of Covariate Adaptive Randomized Designs: The Fractional Brownian Motion Approach

2021

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Conditional power based on classical Brownian motion (BM) has been widely used in sequential monitoring of clinical trials, including those with the covariate adaptive randomization design (CAR). Due to some uncontrollable factors, the sequential test statistics under CAR procedures may not satisfy the independent increment property of BM. We confirm the invalidation of BM when the error terms in the linear model with CAR design are not independent and identically distributed. To incorporate the possible correlation structure of the increment of the test statistic, we utilize the fractional Brownian motion (FBM). We conducted a comparative study of the conditional power under BM and FBM. It was found that the conditional power under FBM assumption was mostly higher than that under BM assumption when the Hurst exponent was greater than 0.5.

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On the Integral of the Fractional Brownian Motion and Some Pseudo-Fractional Gaussian Processes

Cited by 11 publications

References 26 publications

Methods for Mathematical Analysis of Simulated and Real Fractal Processes with Application in Cardiology

Methods for Mathematical Analysis of Simulated and Real Fractal Processes with Application in Cardiology

Efficient or Fractal Market Hypothesis? A Stock Indexes Modelling Using Geometric Brownian Motion and Geometric Fractional Brownian Motion

Estimating Conditional Power for Sequential Monitoring of Covariate Adaptive Randomized Designs: The Fractional Brownian Motion Approach

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