References
- Meerschaert, M.M., Sabzikar, F. (2013). Tempered fractional Brownian motion. Stat. Probab. Lett. 83(10):2269–2275. DOI: 10.1016/j.spl.2013.06.016.
- Sabzikar, F., Meerschaert, M.M., Chen, J.H. (2015). Tempered fractional calculus. J. Comput. Phys. 293:14–28. DOI: 10.1016/j.jcp.2014.04.024.
- Embrechts, P., Maejima, M. (2002). Selfsimilar Processes, Princeton Series in Applied Mathematics. Princeton, NJ: Princeton University Press.
- Giraitis, L., Kokoszka, P., Leipus, R. (2000). Stationary ARCH models: dependence structure and central limit theorem. Econom. Theory 16(1):3–22. DOI: 10.1017/S0266466600161018.
- Kolmogoroff, A.N. (1940). Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. Acad. Sci. URSS. 26:115–118.
- Biagini, F., Hu, Y.Z., Øksendal, B., Zhang, T.S. (2008). Stochastic Calculus for Fractional Brownian Motion and Applications. London: Springer.
- HIlle, E. (1926). A class of reciprocal functions. Ann. Math. 27(4):427–464. DOI: 10.2307/1967695.
- Boniece, B.C., Didier, G., Sabzikar, F. (2021). Tempered fractional Brownian motion: wavelet estimation, modeling and testing. Appl. Comput. Harmon. Anal. 51:461–509. DOI: 10.1016/j.acha.2019.11.004.
- Chevillard, L. (2017). Regularized fractional Ornstein-Uhlenbeck processes and their relevance to the modeling of fluid turbulence. Phys. Rev. E. 96(3):033111. DOI: 10.1103/PhysRevE.96.033111.
- Dacorogna, M.M., Müller, U.A., Nagler, R.J., Olsen, R.B., Pictet, O.V. (1993). A geographical model for the daily and weekly seasonal volatility in the foreign exchange market. J. Int. Money Financ. 12(4):413–438. DOI: 10.1016/0261-5606(93)90004-U.
- Granger, C.W.J., Ding, Z.X. (1996). Varieties of long memory models. J. Econometrics. 73(1):61–77. DOI: 10.1016/0304-4076(95)01733-X.
- Ling, S.Q., Li, W.K. (2001). Asymptotic inference for nonstationary fractionally integrated autoregressive moving-average models. Econom. Theory. 17(4):738–764. DOI: 10.1017/S0266466601174049.
- Zhang, X.L., Xiao, W.L. (2017). Arbitrage with fractional Gaussian processes. Phys. A. 471:620–628. DOI: 10.1016/j.physa.2016.12.064.
- Duc, L.H., Garrido-Atienza, M.J., Neuenkirch, A., Schmalfuß, B. (2018). Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in (1/2,1). J. Differ. Equ. 264:1119–1145. DOI: 10.1016/j.jde.2017.09.033.
- Duc, L.H., Hong, P.T., Cong, N.D. (2019). Asymptotic stability for stochastic dissipative systems with a Hölder noise. SIAM J. Control Optim. 57(4):3046–3071. DOI: 10.1137/19M1236527.
- Garrido-Atienza, M.J., Neuenkirch, A., Schmalfuß, B. (2018). Asymptotical stability of differential equations driven by Hölder continuous paths. J. Dyn. Differ. Equ. 30(1):359–377. DOI: 10.1007/s10884-017-9574-6.
- Wang, Y.J., Liu, Y.R., Caraballo, T. (2021). Exponential behavior and upper noise excitation index of solutions to evolution equations with unbounded delay and tempered fractional Brownian motions. J. Evol. Equ. 21(2):1779–1807. DOI: 10.1007/s00028-020-00656-0.
- Yan, L.T., Pei, W.Y., Zhang, Z.Z., College of Information Science and Technology, Donghua University, Shanghai 201620, China. (2019). Exponential stability of SDEs driven by FBM with Markovian switching. Discrete Contin. Dyn. Syst. 39(11):6467–6483. DOI: 10.3934/dcds.2019280.
- Caraballo, T., Garrido-Atienza, M.J., Taniguchi, T. (2011). The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion. Nonlinear Anal. 74(11):3671–3684. DOI: 10.1016/j.na.2011.02.047.
- Jang, J.J., Guo, J.S. (1999). Analysis of maximum wind force for offshore structure design. J. Mar. Sci. Technol. 7:43–51. DOI: 10.51400/2709-6998.2511.
- Li, Y.S., Kareem, A. (1990). ARMA systems in wind engineering. Probab. Eng. Mech. 5(2):49–59. DOI: 10.1016/S0266-8920(08)80001-X.
- Bender, C. (2003). An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter. Stoch. Proc. Appl. 104(1):81–106. DOI: 10.1016/S0304-4149(02)00212-0.
- Thangavelu, S. (1993). Lectures on Hermite and Laguerre Expansions. Princeton, NJ: Princeton University Press.
- Elliott, R.J., Hoek, J.V.D. (2003). A general fractional white noise theory and applications to finance. Math. Finance 13(2):301–330. DOI: 10.1111/1467-9965.00018.
- Chen, Y., Wang, X.D., Deng, W.H. (2017). Localization and ballistic diffusion for the tempered fractional Brownian-Langevin motion. J. Stat. Phys. 169(1):18–37. DOI: 10.1007/s10955-017-1861-4.
- Liemert, A., Sandev, T., Kantz, H. (2017). Generalized Langevin equation with tempered memory kernel. Phys. A. 466:356–369. DOI: 10.1016/j.physa.2016.09.018.
- Sandev, T., Chechkin, A., Kantz, H., Metzler, R. (2015). Diffusion and Fokker-Planck-Smoluchowski equations with generalized memory kernel. Fract. Calc. Appl. Anal. 18(4):1006–1038. DOI: 10.1515/fca-2015-0059.
- Stanislavsky, A., Weron, K., Weron, A. (2008). Diffusion and relaxation controlled by tempered α-stable processes. Phys. Rev. E. 78(5):051106. DOI: 10.1103/PhysRevE.78.051106.
- Wu, X.C., Deng, W.H., Barkai, E. (2016). Tempered fractional Feynman-Kac equation: theory and examples. Phys. Rev. E. 93(3):032151. DOI: 10.1103/PhysRevE.93.032151.
- Davenport, A.G. (1961). The spectrum of horizontal gustiness near the ground in high winds. Q.J. Royal Met. Soc. 87(372):194–211. DOI: 10.1002/qj.49708737208.
- Meerschaert, M.M., Sabzikar, F., Phanikumar, M.S., Zeleke, A. (2014). Tempered fractional time series model for turbulence in geophysical flows. J. Stat. Mech. 2014(9):P09023–5468. DOI: 10.1088/1742-5468/2014/09/P09023.
- Kuo, H.H. (1996). White Noise Distribution Theory. Boca Raton, FL: CRC Press.
- Holden, H., Øksendal, B., Ubøe, J., Zhang, T. (1996). Stochastic Partial Differential Equations, a Modeling, White Noise Functional Analysis. Cambridge, MA: Birkhäuser.
- Mémin, J., Mishura, Y., Valkeila, E. (2001). Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Statist. Probab. Lett. 51(2):197–206. DOI: 10.1016/S0167-7152(00)00157-7.
- Meerschaert, M.M., Sabzikar, F. (2014). Stochastic integration for tempered fractional Brownian motion. Stoch. Proc. Appl. 124(7):2363–2387. DOI: 10.1016/j.spa.2014.03.002.
- Duncan, T.E., Hu, Y.Z., Pasik-Duncan, B. (2000). Stochastic calculus for fractional Brownian motion, I. Theory. SIAM J. Control. Optim. 38(2):582–612. DOI: 10.1137/S036301299834171X.
- Hu, Y.Z. (2018). Itô Stochastic differential equations driven by fractional Brownian motion of Hurst parameter H>1/2. Stochastics 90:720–761. DOI: 10.1080/17442508.2017.1415342.
- Hu, Y.Z. (2017). Analysis on Gaussian Spaces. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd.
- Azmoodeh, E., Mishura, Y., Sabzikar, F. (2022). How does tempering affect the local and global properties of fractional Brownian motion? J. Theor. Probab. 35(1):484–527. DOI: 10.1007/s10959-020-01068-z.
- Sabzikar, F., Surgailis, D. (2018). Tempered fractional Brownian and stable motions of second kind. Statist. Probab. Lett. 132:17–27. DOI: 10.1016/j.spl.2017.08.015.
- Sabzikar, F., Kabala, J., Burnecki, K. (2022). Tempered fractionally integrated process with stable noise as a transient anomalous diffusion model. J. Phys. A: Math. Theor. 55(17):174002. DOI: 10.1088/1751-8121/ac5b92.
- Kabala, J., Burnecki, K., Sabzikar, F. (2021). Tempered linear and non-linear time series models and their application to heavy-tailed solar flare data. Chaos 31(11):113124. DOI: 10.1063/5.0061754.
- Sun, X.X., Guo, F. (2018). On moment estimates and continuity for solutions of SDEs driven by fractional Brownian motions under non-Lipschitz conditions. Statist. Probab. Lett. 132:116–124. DOI: 10.1016/j.spl.2017.09.008.
- Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Berlin, Germany: Springer.