Geodesic
On upper functions for anomalous diffusions governed by time-varying Ornstein–Uhlenbeck process
Teoriâ veroâtnostej i ee primeneniâ, Tome 64 (2019) no. 2, pp. 258-282 Cet article a éte moissonné depuis la source Math-Net.Ru

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We obtain upper functions that serve as almost sure asymptotic upper bounds for a displacement process given by an integrated time-varying Ornstein–Uhlenbeck process. The form of upper functions depends on the characteristics (the stability rate and the diffusion coefficient) of a stochastic linear differential equation. We introduce the notion of anomalous diffusion related to behavior of upper functions and compare the results of diffusion classification (normal diffusion, subdiffusion, and superdiffusion) with those obtained on the basis of mean square displacements.
Keywords: time-varying Ornstein–Uhlenbeck process, upper function, the law of the iterated logarithm.
Mots-clés : anomalous diffusion 
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E. S. Palamarchuk. On upper functions for anomalous diffusions governed by time-varying Ornstein–Uhlenbeck process. Teoriâ veroâtnostej i ee primeneniâ, Tome 64 (2019) no. 2, pp. 258-282. http://geodesic.mathdoc.fr/item/TVP_2019_64_2_a2/

[1] G. A. Pavliotis, Stochastic processes and applications. Diffusion processes, the Fokker–Planck and Langevin equations, Texts Appl. Math., 60, New York, Berlin, 2014, xiv+339 pp. | DOI | MR | Zbl

[2] D. R. Cox, H. D. Miller, The theory of stochastic processes, Chapman and Hall/CRC, London, 2017, x+398 pp. | DOI | MR | Zbl

[3] E. S. Palamarchuk, “An analytic study of the Ornstein–Uhlenbeck process with time-varying coefficients in the modeling of anomalous diffusions”, Autom. Remote Control, 79:2 (2018), 289–299 | DOI | MR | Zbl

[4] E. S. Palamarchuk, “On the generalization of logarithmic upper function for solution of a linear stochastic differential equation with a nonexponentially stable matrix”, Differ. Equ., 54:2 (2018), 193–200 | DOI | DOI | MR | Zbl

[5] I. I. Gikhman, A. V. Skorokhod, Introduction to the theory of random processes, W. B. Saunders Co., Philadelphia, PA–London–Toronto, ON, 1969, xiii+516 pp. | MR | MR | Zbl

[6] T. A. Belkina, Yu. M. Kabanov, E. L. Presman, “On a stochastic optimality of the feedback control in the LQG-problem”, Theory Probab. Appl., 48:4 (2004), 592–603 | DOI | DOI | MR | Zbl

[7] A. V. Bulinskii, A. N. Shiryaev, Teoriya sluchainykh protsessov, Fizmatlit, M., 2003, 400 pp.

[8] W. G. Cumberland, Z. M. Sykes, “Weak convergence of an autoregressive process used in modeling population growth”, J. Appl. Probab., 19:2 (1982), 450–455 | DOI | MR | Zbl

[9] S. L. Heston, “A closed-form solution for options with stochastic volatility with applications to bond and currency options”, Rev. Financ. Stud., 6:2 (1993), 327–343 | DOI | Zbl

[10] B. Bassan, S. Terzi, “Parameter estimation in systems of differential equations, based on three-way data arrays and random time changes”, Appl. Stochastic Models Data Anal., 12:2 (1996), 63–73 | 3.0.CO;2-F class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[11] S. Ditlevsen, M. Sørensen, “Inference for observations of integrated diffusion processes”, Scand. J. Statist., 31:3 (2004), 417–429 | DOI | MR | Zbl

[12] M. Abundo, E. Pirozzi, “Integrated stationary Ornstein–Uhlenbeck process, and double integral processes”, Phys. A, 494 (2018), 265–275 | DOI | MR

[13] E. S. Palamarchuk, “Optimization of the superstable linear stochastic system applied to the model with extremely impatient agents”, Autom. Remote Control, 79:3 (2018), 439–450 | DOI | MR | Zbl

[14] H. Cramér, M. R. Leadbetter, Stationary and related stochastic processes. Sample function properties and their applications, John Wiley Sons, Inc., New York–London–Sydney, 1967, xii+348 pp. | MR | Zbl | Zbl

[15] T. A. Belkina, E. S. Palamarchuk, “On stochastic optimality for a linear controller with attenuating disturbances”, Autom. Remote Control, 74:4 (2013), 628–641 | DOI | MR | Zbl

[16] E. S. Palamarchuk, “Analysis of the asymptotic behavior of the solution to a linear stochastic differential equation with subexponentially stable matrix and its application to a control problem”, Theory Probab. Appl., 62:4 (2018), 522–533 | DOI | DOI | MR | Zbl

[17] Jia-gang Wang, “A law of the iterated logarithm for stochastic integrals”, Stochastic Process. Appl., 47:2 (1993), 215–228 | DOI | MR | Zbl

[18] E. S. Palamarchuk, “Stabilization of linear stochastic systems with a discount: modeling and estimation of the long-term effects from the application of optimal control strategies”, Math. Models Comput. Simul., 7:4 (2015), 381–388 | DOI | MR | Zbl

[19] H. Safdari, A. G. Cherstvy, A. V. Chechkin, A. Bodrova, R. Metzler, “Aging underdamped scaled Brownian motion: ensemble- and time-averaged particle displacements, nonergodicity, and the failure of the overdamping approximation”, Phys. Rev. E, 95:1 (2017), 012120 | DOI

[20] T. Narumi, M. Suzuki, Y. Hidaka, T. Asai, S. Kai, “Active Brownian motion in threshold distribution of a Coulomb blockade model”, Phys. Rev. E, 84:5 (2011), 051137 | DOI

[21] P. L. Smith, C. R. L. McKenzie, “Diffusive information accumulation by minimal recurrent neural models of decision making”, Neural Comput., 23:8 (2011), 2000–2031 | DOI | MR