1. bookVolume 29 (2021): Edizione 3 (November 2021)
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Rivista
eISSN
1844-0835
Prima pubblicazione
17 May 2013
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1 volta all'anno
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access type Accesso libero

An approximate Taylor method for Stochastic Functional Differential Equations via polynomial condition

Pubblicato online: 23 Nov 2021
Volume & Edizione: Volume 29 (2021) - Edizione 3 (November 2021)
Pagine: 105 - 133
Ricevuto: 22 Mar 2021
Accettato: 30 Apr 2021
Dettagli della rivista
License
Formato
Rivista
eISSN
1844-0835
Prima pubblicazione
17 May 2013
Frequenza di pubblicazione
1 volta all'anno
Lingue
Inglese
Abstract

The subject of this paper is an analytic approximate method for a class of stochastic functional differential equations with coefficients that do not necessarily satisfy the Lipschitz condition nor linear growth condition but they satisfy some polynomial conditions. Also, equations from the observed class have unique solutions with bounded moments. Approximate equations are defined on partitions of the time interval and their drift and diffusion coefficients are Taylor approximations of the coefficients of the initial equation. Taylor approximations require Fréchet derivatives since the coefficients of the initial equation are functionals. The main results of this paper are the Lp and almost sure convergence of the sequence of the approximate solutions to the exact solution of the initial equation. An example that illustrates the theoretical results and contains the proof of the existence, uniqueness and moment boundedness of the approximate solution is displayed.

Keywords

MSC 2010

[1] M. A. Atalla, Finite-difference approximations for stochastic differential equations, Probabilistic Methods for the Investigation of Systems with an Infinite number of Degrees of freedom, Inst. of Math. Acad. of Science USSR, Kiev, (1986) 11–16 (in Russian). Search in Google Scholar

[2] M. A. Atalla, On one approximating method for stochastic differential equations, Asymptotic Methods for the Theory of Stochastic processes, Inst. of Math. Acad. of Science USSR, Kiev, (1987) 15–21 (in Russian). Search in Google Scholar

[3] A. Bahar, X. Mao, Stochastic delay population dynamics, International J. Pure Appl. Math., 11 (2004) 377–400. Search in Google Scholar

[4] L. Collatz, “Functional analysis and numerical mathematics”, Academic Press, New York - San Francisco - London (1966). Search in Google Scholar

[5] T. M. Flett, “Differential analysis”, Cambridge University Press, Cambridge (1980).10.1017/CBO9780511897191 Search in Google Scholar

[6] Y. Guo, W. Zhao, X. Ding, Input-to-state stability for stochastic multi-group models with multy-disperal and time-varying delay, App. Math. Comp., 343 (2019) 114–127. Search in Google Scholar

[7] D. J. Higham, X. Mao, A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (3) (2002) 1041–1063.10.1137/S0036142901389530 Search in Google Scholar

[8] M. Jansen, P. Pfaffelhuber, Stochastic gene expression with delay, J. Theor. Biol., 364 (2015) 355–363. Search in Google Scholar

[9] M. Jovanović, M. Krstić, The influence of time-dependent delay on behavior of stochastic population model with the Allee effect, App. Math. Modell., 39(2) (2015) 733–746.10.1016/j.apm.2014.06.019 Search in Google Scholar

[10] V.B. Kolmanovskii, V.R. Nosov, Stability of Functional Differential Equations, Academic Press, 1986. Search in Google Scholar

[11] M. Krstić, The effect of stochastic perturbation on a nonlinear delay malaria epidemic model, Math. Comput. Simulat., 82(4) 558–569.10.1016/j.matcom.2011.09.003 Search in Google Scholar

[12] M. Liu, C. Bai, Y. Jin, Population dynamical behavior of a two-predator one-prey stochastic model with time delay, Discrete Contin. Dyn. Sys. A, 37 (5) (2017) 2513–2538.10.3934/dcds.2017108 Search in Google Scholar

[13] Q. Liu, D. Jiang, T. Hayat, A. Alsaedi, Dynamics of a stochastic SIR epidemic model with distributed delay and degenerate diffusion, J. Frankl. Inst., 356 (13) (2019) 7347–7370.10.1016/j.jfranklin.2019.06.030 Search in Google Scholar

[14] X. Mao, “Stochastic differential equations and applications”, Horwood Publishing, Chichester, (2008).10.1533/9780857099402 Search in Google Scholar

[15] X. Mao, S. Sabanis, Numerical solutions of stochastic differential delay equations under local Lipschitz condition, J. Comput. Appl. Math., 151 (2003) 215–227. Search in Google Scholar

[16] M. Milošević, M. Jovanović, S. Janković, An approximate method via Taylor series for stochastic functional differential equations, J. Math. Anal. Appl., 363 (2010) 128–137. Search in Google Scholar

[17] M. Obradović, M. Milošević, Stability of a class of neutral stochastic differential equations with unbounded delay and Markovian switching and the Euler?Maruyama method, J. Comput. Appl. Math., 244–266 (2017).10.1016/j.cam.2016.06.038 Search in Google Scholar

[18] J. A. Oguntuase, On integral inequalities of Gronwall–Bellman–Bihari type in several variables, J. Ineq. Pure Appl. Math., 1 (2) (2000). Search in Google Scholar

[19] B. Øksendal, A. Sulem, T. Zhang, Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations, Adv. Appl. Probab., 43(2) (2011) 572–596.10.1239/aap/1308662493 Search in Google Scholar

[20] M. Song, L. Hu, X. Mao, L. Zhang, Khasminskii-type theorems for stochastic functional differential equations, Discrete and continuous dynamical systems, series B, 18 (6) (2013) 1697–1714.10.3934/dcdsb.2013.18.1697 Search in Google Scholar

[21] B. Tojtovska, S. Janković, General decay stability analysis of impulsive neural networks with mixed time delays, Neurocomputing, 142 (2014) 438–446. Search in Google Scholar

[22] M. Vasilova, Asymptotic behavior of a stochastic Gilpin–Ayala predator-prey system with time-dependent delay, Math. Comp. Model., 57 (3-4) (2013) 764–781.10.1016/j.mcm.2012.09.002 Search in Google Scholar

[23] X. Wang, J. Yu, C. Li, H. Wang, T. Huang, J. Huang, Robust stability of stochastic fuzzy delayed neural networks with impulsive time window, Neural Networks, 67 (2015) 84–91. Search in Google Scholar

[24] F. Wu, X. Mao, Numerical solutions of neutral stochastic functional differential equations, SIAM J. Numer. Anal. 46 (2008) 1821–1841.10.1137/070697021 Search in Google Scholar

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