Geodesic
Existence and uniqueness of solutions to stochastic fractional differential equations in multiple time scales
Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 141, pp. 51-59 Cet article a éte moissonné depuis la source Math-Net.Ru

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A novel class of nonlinear stochastic fractional differential equations with delay and the Jumarie and Itô differentials is introduced in the paper. The aim of the study is to prove existence and uniqueness of solutions to these equations. The main results of the paper generalise some previous findings made for the non-delay and three-scale equations under additional restrictions on the fractional order of the Jumarie differentials, which are removed in our analysis. The techniques used in the paper are based on the properties of the singular integral operators in specially designed spaces of stochastic processes, the representation of delay equations as functional differential equations as well as Picard's iterative method.
Keywords: Jumarie derivative, Brownian motion
Mots-clés : multi-time scales. 
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A. V. Ponosov. Existence and uniqueness of solutions to stochastic fractional differential equations in multiple time scales. Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 141, pp. 51-59. http://geodesic.mathdoc.fr/item/VTAMU_2023_28_141_a4/

[1] B. N. Sadovskii, “A fixed-point principle”, Funct. Anal. Appl., 1:2 (1967), 151–153 | MR

[2] J. Jacod, J. Memin, “Existence of weak solutions for stochastic differential equations with driving semimartingales”, Stochastics, 4 (1981), 317–337 | DOI | MR

[3] A. Ponosov, “Fixed point method in the theory of stochastic differential equations”, Soviet Math. Doklady, 37:2 (1988), 426–429 | MR

[4] J. Appell, P. P. Zabreiko, Nonlinear Superposition Operators, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2008, 311 pp. | MR

[5] A. Ponosov, “On the Nemytskii conjecture”, Soviet Math. Doklady, 34:1 (1987), 231–233 | MR

[6] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing ltd., Chichester, 1997, 366 pp. | MR

[7] B. Øksendal, Stochastic Differential Equations, Universitext, Springer-Verlag Berlin Heidelberg, Berlin, 2013, 379 pp. | MR

[8] A. Ponosov, “Local operators and stochastic differential equations”, Functional Differential Equations, 4:1–2 (1997), 73–89 | MR

[9] G. Di Nunno, B. Øksendal, F. Proske, Malliavin Calculus for Lévy Processes with Applications to Finance, Universitext, Springer–Verlag Berlin Heidelberg, Berlin, 2009, 418 pp. | MR

[10] D. H. Wagner, “Survey of measurable selection theorems”, SIAM J. Control and Optimization, 15:5 (1977), 859–903 | DOI | MR

[11] I. V. Shragin, “Abstract Nemytskii operators are locally defined operators”, Soviet Math. Doklady, 17:2 (1976), 354–357 | MR

[12] A. Ponosov, E. Stepanov, “Atomic operators, random dynamical systems and invariant measures”, St. Petersburg Math. J., 26 (2015), 607–642 | DOI | MR

[13] S. G. Krantz, Function Theory of Several Complex Variables: Second Edition, v. 340, AMS Chelsea Publishing, Providence, 1992 | MR

[14] Hu Sze-Tsen, Theory of Retracts, Wayne State University Press, Detroit, 1965, 234 pp. | MR