Fractional Brownian Motions, Fractional Noises and Applications

Previous article Next article Fractional Brownian Motions, Fractional Noises and ApplicationsBenoit B. Mandelbrot and John W. Van NessBenoit B. Mandelbrot and John W. Van Nesshttps: //doi. org/10. 1137/1010093PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout1 I. Adelman, Long cycles—fact or artifact? , Amer. Economic Rev. , 60 (1965), 444–463 Google Scholar2 William Feller, The asymptotic distribution of the range of sums of independent random variables, Ann. Math. Statistics, 22 (1951), 427–432 MR0042626 0043. 34201 CrossrefISIGoogle Scholar3 I. M. Gel'fand and, N. Ya. Vilenkin, Generalized functions. Vol. 4, Academic Press Harcourt Brace Jovanovich Publishers, New York, 1964 1977xiv+384 MR0435834 Google Scholar4 C. W. J. Granger, The typical spectral shape of an economic variable, Econometrica, 34 (1966), 150–161 CrossrefISIGoogle Scholar5 G. A. Hunt, Random Fourier transforms, Trans. Amer. Math. Soc. , 71 (1951), 38–69 MR0051340 0043. 30601 CrossrefISIGoogle Scholar6 H. E. Hurst, , R. P. Black and, Y. M. Sinaika, Long Term Storage in Reservoirs. An Experimental Study, Constable, London, 1965 Google Scholar7 A. N. Kolmogoroff, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum, C. R. (Doklady) Acad. Sci. URSS (N. S. ), 26 (1940), 115–118 MR0003441 0022. 36001 Google Scholar8 John Lamperti, Semi-stable stochastic processes, Trans. Amer. Math. Soc. , 104 (1962), 62–78 MR0138128 0286. 60017 CrossrefGoogle Scholar9 Paul Lévy, Random functions: General theory with special reference to Laplacian random functions, Univ. California Publ. Statist. , 1 (1953), 331–390 MR0055607 0052. 14402 Google Scholar10 Michel Loève, Probability theory, 2nd ed. The University Series in Higher Mathematics. D. Van Nostrand Co. , Inc. , Princeton, N. J. -Toronto-New York-London, 1960xvi+685 MR0123342 0095. 12201 Google Scholar11 Benoı⁁t Mandelbrot, Une classe processus stochastiques homothétiques à soi; application à la loi climatologique H. E. Hurst, C. R. Acad. Sci. Paris, 260 (1965), 3274–3277 MR0176521 0127. 09501 Google Scholar12 B. Mandelbrot, Self-similar error-clusters in communication systems and the concept of conditional stationarity, IEEE Trans. Comet. Tech. , COM-13 (1965), 71–90 10. 1109/TCOM. 1965. 1089090 CrossrefISIGoogle Scholar13 B. Mandelbrot, Noises with an l/f spectrum, a bridge between direct current and white noise, IEEE Trans. Information Theory, IT-13 (1967), 289–298 10. 1109/TIT. 1967. 1053992 0148. 40507 CrossrefISIGoogle Scholar14 Benoit Mandelbrot, Sporadic random functions and conditional spectral analysis: Self-similar examples and limitsProc. Fifth Berkeley Sympos. Mathematical Statistics and Probability (Berkeley, Calif. , 1965/66), Vol. III: Physical Sciences, Univ. California Press, Berkeley, Calif. , 1967, 155–179 MR0224243 0189. 18302 Google Scholar15 B. Mandelbrot and, J. R. Wallis, Noah, Joseph and operational hydrology, Water Resources Research, to appear Google Scholar16 B. Mandelbrot and, J. R. Wallis, Computer experiments with fractional Gaussian noise, Water Resources Research, to appear Google Scholar17 B. Mandelbrot and, J. R. Wallis, Some long run properties of geophysical records, Water Resources Research, to appear. Google Scholar18 G. Maruyama, The harmonic analysis of stationary stochastic processes, Mem. Fac. Sci. Kyusyu Univ. A. , 4 (1949), 45–106 MR0032127 0045. 40602 CrossrefGoogle Scholar19 M. Rosenblatt, Independence and dependenceProc. 4th Berkeley Sympos. Math. Statist. and Prob. , Vol. II, Univ. California Press, Berkeley, Calif. , 1961, 431–443 MR0133863 0105. 11802 Google Scholar20 G. I. Taylor, Statistical theory of turbulence, Proc. Roy. Sac. Ser. A, 151 (1935), 421–478 CrossrefGoogle Scholar21 H. Weyl, Bemerkungen zum Begriff der Differential-Quotenten gebrochener Ordnung, Vierteljschr. Naturforsch. Ges. Zürich, 62 (1967), 296–302 Google Scholar22 A. M. Yaglom, Correlation theory of processes with random stationary nth increments, Amer. Math. Soc. Transl. (2), 8 (1958), 87–141 MR0091554 0080. 34903 CrossrefGoogle Scholar23 A. M. Yaglom, L. M. LeCam and, J. Neyman, Stationary Gaussian processes satisfying the strong mixing condition and best predictable functionalsBernoulli-Bayes-Laplace Anniversary Volume, Springer-Verlag, New York, 1965, 241–252 0142. 13904 CrossrefGoogle Scholar24 A. M. Yaglom, Outline of some topics in linear extrapolation of stationary random processesProc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif. , 1965/66), Vol. II: Contributions to Probability Theory, Part 1, Univ. California Press, Berkeley, Calif. , 1967, 259–278 MR0215358 0189. 18401 Google Scholar25 Antoni Zygmund, Trigonometrical series, Chelsea Publishing Co. , New York, 1952vi+329, 2nd ed. MR0076084 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Mean square stability of stochastic theta method for stochastic differential equations driven by fractional Brownian motionJournal of Computational and Applied Mathematics, Vol. 420 | 1 Mar 2023 Cross Ref ARMA–GARCH model with fractional generalized hyperbolic innovationsFinancial Innovation, Vol. 8, No. 1 | 15 May 2022 Cross Ref Fokker–Planck equation of the fractional Brownian motionInternational Journal of Non-Linear Mechanics, Vol. 147 | 1 Dec 2022 Cross Ref A generative model for fBm with deep ReLU neural networksJournal of Complexity, Vol. 73 | 1 Dec 2022 Cross Ref Emergence of non-Fickian transport in truncated pluri-Gaussian permeability fieldsGEM - International Journal on Geomathematics, Vol. 13, No. 1 | 1 October 2022 Cross Ref A comparison of maximum likelihood and absolute moments for the estimation of Hurst exponents in a stationary frameworkCommunications in Nonlinear Science and Numerical Simulation, Vol. 114 | 1 Nov 2022 Cross Ref Multiscaling and rough volatility: An empirical investigationInternational Review of Financial Analysis, Vol. 84 | 1 Nov 2022 Cross Ref Multivariate claim processes with rough intensities: Properties and estimationInsurance: Mathematics and Economics, Vol. 107 | 1 Nov 2022 Cross Ref Exploring the dynamic nonlinear relationship between crude oil price and implied volatility indices: A new perspective from MMV-MFDFAPhysica A: Statistical Mechanics and its Applications, Vol. 603 | 1 Oct 2022 Cross Ref On the model of random walk with multiple memory structurePhysica A: Statistical Mechanics and its Applications, Vol. 603 | 1 Oct 2022 Cross Ref Self-similarity and response of fractional differential equations under white noise inputProbabilistic Engineering Mechanics, Vol. 70 | 1 Oct 2022 Cross Ref The truncated Euler–Maruyama method for CIR model driven by fractional Brownian motionStatistics & Probability Letters, Vol. 189 | 1 Oct 2022 Cross Ref Discrete scaling and criticality in a chain of adaptive excitable integratorsChaos, Solitons & Fractals, Vol. 163 | 1 Oct 2022 Cross Ref Novel Neuron-like Procedure of Weak Signal Detection against the Non-Stationary Noise Background with Application to Underwater SoundRemote Sensing, Vol. 14, No. 19 | 29 September 2022 Cross Ref Chaos Analysis Framework: How to Safely Identify and Quantify Time-Series DynamicsNonlinear Systems - Recent Developments and Advances Working Title | 22 September 2022 Cross Ref Asymptotic of the running maximum distribution of a Gaussian BridgeStochastic Analysis and Applications, Vol. 28 | 19 September 2022 Cross Ref Wavelet eigenvalue regression in high dimensionsStatistical Inference for Stochastic Processes, Vol. 1 | 18 September 2022 Cross Ref Data-Driven Modeling of Non-Markovian Noise in Semiconductor Lasers2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD) | 12 Sep 2022 Cross Ref Continuous wavelet estimation for multivariate fractional Brownian motionPakistan Journal of Statistics and Operation Research | 10 September 2022 Cross Ref Persistence in daily returns of stocks with highest market capitalization in the Indian marketDigital Finance, Vol. 503 | 9 September 2022 Cross Ref Bayesian semiparametric long memory models for discretized event dataThe Annals of Applied Statistics, Vol. 16, No. 3 | 1 Sep 2022 Cross Ref Scale invariance in fNIRS as a measurement of cognitive loadCortex, Vol. 154 | 1 Sep 2022 Cross Ref Efficient calculation of fractal properties via the Higuchi methodNonlinear Dynamics, Vol. 109, No. 4 | 23 June 2022 Cross Ref N-Fold Compound Option Fuzzy Pricing Based on the Fractional Brownian MotionInternational Journal of Fuzzy Systems, Vol. 24, No. 6 | 9 May 2022 Cross Ref On the nature and propagation of errors in roughness parameters obtained from spectral analysis of atomic force microscopy topographic imagesJournal of Vacuum Science & Technology A, Vol. 40, No. 5 | 1 Sep 2022 Cross Ref Resemblance of the power-law scaling behavior of a non-Markovian and nonlinear point processesChaos, Solitons & Fractals, Vol. 162 | 1 Sep 2022 Cross Ref Study on the Effect of Judgment Excitation Mode to Relieve Driving Fatigue Based on MF-DFABrain Sciences, Vol. 12, No. 9 | 6 September 2022 Cross Ref Fractional Brownian motion with random Hurst exponent: Accelerating diffusion and persistence transitionsChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 32, No. 9 | 1 Sep 2022 Cross Ref How critical is brain criticality? Trends in Neurosciences, Vol. 21 | 1 Sep 2022 Cross Ref Estimation of hurst exponent for sequential monitoring of clinical trials with covariate adaptive randomizationContemporary Clinical Trials, Vol. 120 | 1 Sep 2022 Cross Ref A GMM approach to estimate the roughness of stochastic volatilityJournal of Econometrics, Vol. 57 | 1 Sep 2022 Cross Ref Frequency–frequency correlations of single-trajectory spectral densities of Gaussian processesNew Journal of Physics, Vol. 24, No. 9 | 27 September 2022 Cross Ref Dispersion heterogeneous recurrence analysis and its use on fault detectionCommunications in Nonlinear Science and Numerical Simulation, Vol. 3 | 1 Sep 2022 Cross Ref UNIFORM CONTINUITY OF FRACTIONAL ORDER INTEGRAL OF FRACTAL INTERPOLATION FUNCTIONFractals, Vol. 30, No. 06 | 6 June 2022 Cross Ref Order pattern recurrence for the analysis of complex systemsPhysica A: Statistical Mechanics and its Applications, Vol. 65 | 1 Sep 2022 Cross Ref Multifractal detrended cross-correlation analysis of radioactivity borne radon, thoron and meteorological time seriesPhysica A: Statistical Mechanics and its Applications, Vol. 10 | 1 Sep 2022 Cross Ref Recovering mixtures of fast-diffusing states from short single-particle trajectorieseLife, Vol. 11 | 30 August 2022 Cross Ref Shadow Price Approximation for the Fractional Black Scholes ModelInternational Journal of Mathematics and Mathematical Sciences, Vol. 2022 | 29 Aug 2022 Cross Ref Roughness and intermittency within metropolitan regions - Application in three French conurbationsEnvironment and Planning B: Urban Analytics and City Science, Vol. 215 | 27 August 2022 Cross Ref Understanding the structure of cognitive noisePLOS Computational Biology, Vol. 18, No. 8 | 17 August 2022 Cross Ref Edgeworth expansion of the t-statistic of the whittle MLE for linear regression processes with long-memory disturbancesCommunications in Statistics - Theory and Methods, Vol. 3 | 16 August 2022 Cross Ref Anomalous diffusion, aging, and nonergodicity of scaled Brownian motion with fractional Gaussian noise: overview of related experimental observations and modelsPhysical Chemistry Chemical Physics, Vol. 24, No. 31 | 1 January 2022 Cross Ref UV Slopes at All Redshifts Are Consistent with H = 1 Stochastic Star Formation HistoriesResearch Notes of the AAS, Vol. 6, No. 8 | 10 August 2022 Cross Ref A stochastic model to reproduce the star formation history of individual galaxies in hydrodynamic simulationsMonthly Notices of the Royal Astronomical Society, Vol. 515, No. 3 | 13 July 2022 Cross Ref Forecasting with fractional Brownian motion: a financial perspectiveQuantitative Finance, Vol. 22, No. 8 | 20 May 2022 Cross Ref Optimal control of non-instantaneous impulsive second-order stochastic McKean–Vlasov evolution system with Clarke subdifferentialInternational Journal of Nonlinear Sciences and Numerical Simulation, Vol. 0, No. 0 | 3 August 2022 Cross Ref Pairs trading with fractional Ornstein–Uhlenbeck spread modelApplied Economics, Vol. 10 | 3 August 2022 Cross Ref Stochastic particle unbinding modulates growth dynamics and size of transcription factor condensates in living cellsProceedings of the National Academy of Sciences, Vol. 119, No. 31 | 26 July 2022 Cross Ref Option pricing based on a type of fuzzy processJournal of Ambient Intelligence and Humanized Computing, Vol. 13, No. 8 | 26 June 2021 Cross Ref Entropic Compressibility of Lévy ProcessesIEEE Transactions on Information Theory, Vol. 68, No. 8 | 1 Aug 2022 Cross Ref Comment on “A computational technique to classify several fractional Brownian motion processes” Chaos Solitons and Fractals 150 (2021) 111152Chaos, Solitons & Fractals, Vol. 161 | 1 Aug 2022 Cross Ref Fractional Gaussian fields on the Sierpiński Gasket and related fractalsJournal d'Analyse Mathématique, Vol. 146, No. 2 | 11 July 2022 Cross Ref Multifractal Characterization and Modeling of Blood Pressure SignalsAlgorithms, Vol. 15, No. 8 | 26 July 2022 Cross Ref Robustness of Detrended Cross-correlation Analysis Method Under Outliers ObservationsFluctuation and Noise Letters, Vol. 21, No. 04 | 21 April 2022 Cross Ref Entropy fluctuation and correlation transfer in tunable discrete-time quantum walk with fractional Gaussian noisePhysical Review E, Vol. 106, No. 2 | 15 August 2022 Cross Ref Poincaré Plot Nonextensive Distribution Entropy: A New Method for Electroencephalography (EEG) Time SeriesSensors, Vol. 22, No. 16 | 21 August 2022 Cross Ref A Model Based on Fractional Brownian Motion for Temperature Fluctuation in the Campi Flegrei CalderaFractal and Fractional, Vol. 6, No. 8 | 30 July 2022 Cross Ref Acceleration scaling and stochastic dynamics of a fluid particle in turbulencePhysical Review Fluids, Vol. 7, No. 8 | 24 August 2022 Cross Ref Multifractional Brownian motion characterization based on Hurst exponent estimation and statistical learningChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 32, No. 8 | 1 Aug 2022 Cross Ref Stability analysis of the Iraqi GNSS stationsJournal of Applied Geodesy, Vol. 16, No. 3 | 31 March 2022 Cross Ref Multifractal test for nonlinearity of interactions across scales in time seriesBehavior Research Methods, Vol. 61 | 19 July 2022 Cross Ref Rolling Bearing Fault Diagnosis Based on MFDFA-SPS and ELMMathematical Problems in Engineering, Vol. 2022 | 18 Jul 2022 Cross Ref Encoding Protest Duration In An Agent-Based Model As Characteristic Phase Transitions2022 Annual Modeling and Simulation Conference (ANNSIM) | 18 Jul 2022 Cross Ref Decomposing the effect of anomalous diffusion enables direct calculation of the Hurst exponent and model classification for single random pathsJournal of Physics A: Mathematical and Theoretical, Vol. 55, No. 27 | 13 June 2022 Cross Ref Classification of stochastic processes by convolutional neural networksJournal of Physics A: Mathematical and Theoretical, Vol. 55, No. 27 | 24 June 2022 Cross Ref A generalized two‐factor square‐root framework for modeling occurrences of natural catastrophesJournal of Forecasting, Vol. 4 | 6 July 2022 Cross Ref The first exit time of fractional Brownian motion from the minimum and maximum parabolic domainsStatistics & Probability Letters, Vol. 186 | 1 Jul 2022 Cross Ref Fractional stochastic Loewner evolution and scaling curvesPhysica A: Statistical Mechanics and its Applications, Vol. 597 | 1 Jul 2022 Cross Ref Stochastic pursuit-evasion curves for foraging dynamicsPhysica A: Statistical Mechanics and its Applications, Vol. 597 | 1 Jul 2022 Cross Ref Power‐type derivatives for rough volatility with jumpsJournal of Futures Markets, Vol. 42, No. 7 | 13 May 2022 Cross Ref Simultaneous Identification of Volatility and Mean-Reverting Parameter for European Option under Fractional CKLS ModelFractal and Fractional, Vol. 6, No. 7 | 21 June 2022 Cross Ref Absence of stationary states and non-Boltzmann distributions of fractional Brownian motion in shallow external potentialsNew Journal of Physics, Vol. 24, No. 7 | 7 July 2022 Cross Ref Sublinear diffusion in the generalized triangle mapPhysical Review E, Vol. 106, No. 1 | 11 July 2022 Cross Ref Multivariate long memory structure in the cryptocurrency market: The impact of COVID-19International Review of Financial Analysis, Vol. 82 | 1 Jul 2022 Cross Ref Unravelling the origins of anomalous diffusion: From molecules to migrating storksPhysical Review Research, Vol. 4, No. 3 | 18 July 2022 Cross Ref Fractality of Borsa Istanbul during the COVID-19 PandemicMathematics, Vol. 10, No. 14 | 18 July 2022 Cross Ref Probing local chromatin dynamics by tracking telomeresBiophysical Journal, Vol. 121, No. 14 | 1 Jul 2022 Cross Ref Are the top six cryptocurrencies efficient? Evidence from time‐varying long memoryInternational Journal of Finance & Economics, Vol. 27, No. 3 | 7 December 2020 Cross Ref The anomalous process in singlet fission kinetic model with time-dependent coefficientPhysica B: Condensed Matter, Vol. 117 | 1 Jul 2022 Cross Ref A Unified Convergence Analysis for the Fractional Diffusion Equation Driven by Fractional Gaussian Noise with Hurst Index H (0, 1) Daxin Nie and Weihua DengSIAM Journal on Numerical Analysis, Vol. 60, No. 3 | 28 June 2022AbstractPDF (498 KB) Elimination of Image Saturation Effects on Multifractal Statistics Using the 2D WTMM in Vol. 13 | 28 June 2022 Cross Ref the relationship between processes with fractional Brownian motion errors of Statistical and Simulation, Vol. 8 | 28 June 2022 Cross Ref under a fractional Brownian motion with Stochastic in and Vol. 2 | 27 June 2022 Cross Ref of rough stochastic volatility models on Journal, Vol. 19 | June 2022 Cross Ref the of anomalous diffusion with the of of Physics A: Mathematical and Theoretical, Vol. 55, No. 24 | 26 May 2022 Cross Ref impact on Evidence from a Hurst A: Statistical Mechanics and its Applications, Vol. | 1 2022 Cross Ref method to the of in fractal and Vol. 10, No. 2 | 18 May 2022 Cross Ref of Using and Statistical in Journal of Science and Transactions of Engineering, Vol. No. 3 | 26 September 2021 Cross Ref The of fractional Brownian Annals of Applied Vol. 32, No. 3 | 1 2022 Cross Ref behavior during the a comparison between and European based on Economic Vol. 12, No. 2 | 20 September 2021 Cross Ref On fractional Lévy motion: and in Applied Vol. No. 2 | 6 June 2022 Cross Ref Time and Persistence in An Using Fractional Journal of Research, Vol. 16, No. 3 | 7 April 2022 Cross Ref A for time fractional with Vol. No. | 2 2021 Cross Ref and Vol. No. 2 | 13 June 2022 Cross Ref Long short time the rough and in and Finance, Vol. No. 1 | 8 2021 Cross Ref properties for a fractional and Applied Analysis, Vol. No. 3 | 13 May 2022 Cross Ref Fractional Lévy motion with for and analysis of Vol. | 1 2022 Cross Ref random in Review E, Vol. No. 6 | 27 June 2022 Cross Ref of of fractional Brownian Review E, Vol. No. 6 | 30 June 2022 Cross Ref of for fractional impulsive differential equations driven by fractional Brownian and Stochastic Vol. 0, No. 0 | 31 May 2022 Cross Ref of fractional stochastic differential equation with and fractional Brownian Vol. | 30 May 2022 Cross Ref the of scales in and Planning B: Urban Analytics and City Science, Vol. 11 | May 2022 Cross Ref the of on of International Conference on and | 23 May 2022 Cross Ref the of in Multivariate by in the Wavelet 2022 - 2022 IEEE International Conference on and Signal | 23 May 2022 Cross Ref estimation for the model driven by the complex fractional Brownian Vol. No. 4 | 28 July 2021 Cross Ref Anomalous diffusion and memory in Computational Biology, Vol. 18, No. 5 | 18 May 2022 Cross Ref OF OF A | 13 May 2022 Cross Ref On the of Vol. 10, No. 9 | 22 April 2022 Cross Ref Convergence of for for with of Computing, Vol. No. 2 | 21 March 2022 Cross Ref Entropy of in a Review E, Vol. No. 5 | 2 May 2022 Cross Ref Analysis of during 26 August Vol. 24, No. 5 | 14 May 2022 Cross Ref stability of stochastic A and Dynamics, Vol. 22, No. | 18 April 2022 Cross Ref behavior in a model with A, Vol. | 1 May 2022 Cross Ref Asymptotic for driven by fractional Brownian motion in the of Mathematical Analysis and Applications, Vol. No. 1 | 1 May 2022 Cross Ref Generalized and rough Ornstein–Uhlenbeck Vol. | 30 April 2022 Cross Ref The the distribution of processes for the estimation of the Sciences Journal, Vol. No. 6 | 12 April 2022 Cross Ref and Application of Some Fractional Driven by Fractional Economics, Vol. 2 | April 2022 Cross Ref for processes and to Vol. No. 4 | 1 March 2022 Cross Ref Stochastic with to fractional processes in of Analysis, Vol. No. 8 | 1 2022 Cross Ref of and in Hurst exponents from spectral analysis of atomic force microscopy topographic Science, Vol. | 1 2022 Cross Ref in Distribution to Stochastic Driven by a Fractional Brownian Journal of Mathematics, Vol. No. 2 | 21 2022 Cross Ref a Under Random and Stochastic and Transactions on Science and Engineering, Vol. No. 2 | 1 2022 Cross Ref A new fractional based on nonlinear structure to systems with and of and Vol. No. | 30 January 2021 Cross Ref and The Transactions on and Systems, Vol. No. 2 | 1 2022 Cross Ref Analysis on with of and Fractional, Vol. 6, No. 4 | 14 April 2022 Cross Ref The of the Fractional and Fractional, Vol. 6, No. 4 | 15 April 2022 Cross Ref A and for complex Review E, Vol. No. 4 | 18 April 2022 Cross Ref of under in Review E, Vol. No. 4 | April 2022 Cross Ref theory of of an Review A, Vol. No. 4 | 26 April 2022 Cross Ref method for model in fractal Solitons & Fractals, Vol. | 1 2022 Cross Ref Time fractional equation with a based fractional Solitons & Fractals, Vol. | 1 2022 Cross Ref stochastic and of with complex in Applied Mechanics and Engineering, Vol. | 1 2022 Cross Ref OF Journal, Vol. No. 2 | 18 August 2022 Cross Ref The of of Fractional by in Vol. 2 | 24 March 2022 Cross Ref Bayesian of scaled fractional Brownian motionJournal of Physics A: Mathematical and | 24 March 2022 Cross Ref Anomalous diffusion: fractional Brownian motion fractional motionJournal of Physics A: Mathematical and Theoretical, Vol. 55, No. 11 | 23 2022 Cross Ref process with noise as a anomalous diffusion of Physics A: Mathematical and | 8 March 2022 Cross Ref by and Analysis and Vol. | 4 March 2022 Cross Ref Stochastic for the stochastic differential equations with & Probability Letters, Vol. | 1 Mar 2022 Cross Ref Multivariate range A: Statistical Mechanics and its Applications, Vol. | 1 Mar 2022 Cross Ref the memory for and time with Vol. No. 3 | 31 January 2022 Cross Ref for the Fractional Ornstein–Uhlenbeck of Vol. No. 1 | 22 October 2020 Cross Ref Properties of a Generalized Fractional Brownian of Vol. No. 1 | 9 January 2021 Cross Ref The for Time of Stochastic Vol. 10, No. 3 | 23 2022 Cross Ref A for Long in Higher Vol. 10, No. 5 | 24 2022 Cross Ref The of time Journal of Statistics, Vol. No. 1 | 14 August 2021 Cross Ref Estimation on of time generalized Dynamics, Vol. No. 4 | 12 January 2022 Cross Ref of the in the of fields by the of the Vol. | 1 Mar 2022 Cross Ref Statistical of nonlinear time and application to Review Research, Vol. 4, No. 1 | 17 March 2022 Cross Ref Price Modeling of under for Vol. 13, No. 3 | 18 March 2022 Cross Ref and in Advances of Physical Sciences, Vol. 06 | 27 April 2022 Cross Ref of of under fractional Brownian motionStatistics & Probability Letters, Vol. | 1 Mar 2022 Cross Ref Fractional and Statistical An of the Indian of Science, Vol. 29 | 24 2022 Cross Ref diffusion with stochastic of Physics A: Mathematical and Theoretical, Vol. 55, No. 7 | 28 January 2022 Cross Ref On the fractional stochastic for random Analysis and Applications, Vol. | 2 2022 Cross Ref of and Systems to