The paper describes the theoretical apparatus and the algorithmic part of a scheme for stochastic examination of systems of linear stochastic delay differential equations perturbed by additive and multiplicative white noises. The presence of an efficient method of analyzing such problems, i.e., a method that can compete with the method of statistical simulation (Monte Carlo), is important both from the theoretical and applied points of view. The systems are substantial for modeling various objects and processes and can be the results of linearization of nonlinear stochastic systems with delay and multiplicative fluctuations. The importance of developing a procedure for calculating just the senior moment functions for the state vectors comes from the presence of multiplicative perturbations implying the inability to restrict the first moment functions as in the case of linear systems with Gaussian white noises as the input. To obtain ODEs for the first and senior moment functions without delays, we apply a new modification of our scheme combining the classical method of steps and extension of the system state space with the addition of the use of multidimensional matrices for the compact recording of the algorithm. The scheme realized in the environment of Wolfram Mathematica software package is demonstrated by an example devoted to an estimation the moment functions until the 4th order for a non-stationary stochastic response of the system with one degree of freedom.
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References
L. Arnold, Stochastic Differential Equations, John Wiley & Sons, New York (1974).
N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Introduction to the Theory of Functional Differential Equations: Methods and Applications, Hindawi Publishing Corporation, New York (2007).
D. Bainov and D. Mishev, Oscillation Theory for Neutral Differential Equations with Delay, Adam Hilger, Bristol (1991).
A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford (2003).
E. Buckwar, “Introduction to the numerical analysis of stochastic delay differential equations,” J. Comput. Appl. Math., 125, No. 1–2, 297–307 (2000).
T. Erneux, Applied Delay Differential Equations, Springer, New York (2009).
E. Fridman, Introduction to Time-Delay Systems: Analysis and Control, Birkhäuser, Basel (2014).
C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer, Berlin (2004).
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York (1993).
A. F. Ivanov, Y. I. Kazmerchuk, and A. V. Swishchuk, “Theory, stochastic stability and applications of stochastic delay differential equations: a survey of recent results,” Differ. Equ. Dyn. Syst., 11, No. 1, 55–115 (2003).
P. Krée and C. Soize, Mathematics of Random Phenomena: Random Vibrations of Mechanical Structures, D. Reidel Publishing Company, Dordrecht (1986).
U. Küchler and E. Platen, “Strong discrete time approximation of stochastic differential equations with time delay,” Math. Comput. Simul., 5, No. 1–3, 189–205 (2000).
H. Kushner, Numerical Methods for Controlled Stochastic Delay Systems, Birkhäuser, Boston (2008).
D. F. Kuznetsov, Stochastic Differental Equations: Theory and Practice of Numerical Solution, Izdatelstvo Sankt-Peterburgskogo Politekhnicheskogo Universiteta, Saint Petersburg, Russia (2010).
M. Lakshmanan and D. Senthilkumar, Dynamics of Nonlinear Time-Delay Systems, Springer, Berlin–Heidelberg (2010).
V. Lakshmikantham and M. Rao, Theory of Integro-Differential Equations, Gordon and Breach Science Publishers, Reading, UK (1995).
A. Longtin, “Stochastic delay-differential equations,” in: Complex Time-Delay Systems. Understanding Complex Systems, F. Atay (ed.), Springer, Berlin (2010), pp. 177–195.
V. V. Malanin and I. E. Poloskov, Methods and Practice of Analysis for Random Processes in Dynamical Systems, Regulyarnaya i Khaoticheskaya Dinamika, Ijevsk, Russia (2005).
X. Mao, Stochastic Differential Equations and Applications, Woodhead Publishing Limited, Cambridge–Oxford, UK (2011).
G. N. Milstein and M. V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer, Berlin–Heidelberg (2004).
S. E. A. Mohammed, Retarded Functional Differential Equations: A Global Point of View, Pitman Publishing, London (1978).
S. E. A. Mohammed, Stochastic Functional Differential Equations, Pitman Advanced Publishing Program, Boston–London (1984).
A. D. Myshkis, “General theory of differential equations with retarded argument,” Russ. Math. Surv., 4, No. 5 (33), 99–141 (1949).
A. D. Myshkis, Linear Differential Equations with Delayed Argument, GITTL, Moscow (1951).
W. Padgett and C. Tsokos, “Stochastic integro-differential equations of Volterra type,” SIAM J. Appl. Math., 23, No. 4, 499–512 (1972).
E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer, Heidelberg (2010).
I. Poloskov, “Phase space extension in the analysis of differential-difference systems with random input,” Autom. Remote Control, 63, No. 9, 1426–1438 (2002).
I. Poloskov, “Symbolic-numeric algorithms for analysis of stochastic systems with different forms of aftereffect,” PAMM, 7, No. 1, 2080011–2080012 (2007).
I. Poloskov, “Numerical and analytical methods of study of stochastic systems with delay,” J. Math. Sci., 230, No. 5, 746–750 (2018).
V. Rubanik, Oscillations of Complex Quasi-Linear Delay Systems, Universitetskoe, Minsk (1985).
L. Shampine, I. Gladwell, and S. Thompson, Solving ODEs with MATLAB, Cambridge University Press, Cambridge (2003).
H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York (2011).
C. Soize, Uncertainty Quantification: an Accelerated Course with Advanced Applications in Computational Engineering, Springer, Cham (2017).
N. P. Sokolov, Introduction to the Theory of Multidimensional Matrices, Naukova Dumka, Kiev (1972).
V. Tikhonov and M. Mironov, Markov Processes, Sovetskoe Radio, Moscow (1977).
E. F. Tsar’kov, “Systems of stochastic delay differential equations,” Latvijas PSR Zinātņu akadēmijas izdevums, Fisikas un Tehnisko Zinātçu Sérija, 2, 57–64 (1966).
E. F. Tsar’kov, Random Perturbations of Functional-Differential Equations, Zinātne, Rīga (1989).
Z. Wang, X. Li, and J. Lei, “Second moment boundedness of linear stochastic delay differential equations,” Discret. Contin. Dyn. Syst.–B, 19, No. 9, 2963–2991 (2014).
J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York (1996).
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Proceedings of the XXXV International Seminar on Stability Problems for Stochastic Models, Perm, Russia, September 24–28, 2018. Part I.
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Poloskov, I.E. New Scheme for Estimation of the First and Senior Moment Functions for the Response of Linear Delay Differential System Excited by Additive and Multiplicative Noises. J Math Sci 246, 525–539 (2020). https://doi.org/10.1007/s10958-020-04757-6
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DOI: https://doi.org/10.1007/s10958-020-04757-6