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Russia Perm
Section Mathematics
Title Stochastic differential equations with random delays in the form of discrete Markov chains
Author(-s) Poloskov I.E.a
Affiliations Perm State National Research Universitya
Abstract The paper provides an overview of the problems that lead to a necessity for analyzing models of linear and nonlinear dynamic systems in the form of stochastic differential equations with random delays of various types as well as some well-known methods for solving these problems. In addition, the author proposes some new approaches to the approximate analysis of linear and nonlinear stochastic dynamic systems. Changes of delays in these systems are governed by discrete Markov chains with continuous time. The proposed techniques for the analysis of systems are based on a combination of the classical steps method, an extension of the state space of a stochastic system under examination, and the method of statistical modeling (Monte Carlo). In this case the techniques allow to simplify the task and to transfer the source equations to systems of stochastic differential equations without delay. Moreover, for the case of linear systems the author has obtained a closed sequence of systems with increasing dimensions of ordinary differential equations satisfied by the functions of conditional expectations and covariances for the state vector. The above scheme is demonstrated by the example of a second-order stochastic system. Changes of the delay in this system are controlled by the Markov chain with five states. All calculations and graphics were performed in the environment of the mathematical package Mathematica by means of a program written in the source language of the package.
Keywords stochastic dynamic system, random delay, modeling, state vector, transition process
UDC 519.21, 004.94
MSC 65C30, 60H35, 68U20
DOI 10.20537/vm150407
Received 20 August 2015
Language Russian
Citation Poloskov I.E. Stochastic differential equations with random delays in the form of discrete Markov chains, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 4, pp. 501-516.
  1. Azbelev N.V., Maksimov V.P., Rakhmatullina L.F. Introduction to the theory of functional differential equations: Methods and applications, New York: Hindawi Publishing Corporation, 2007, 318 p. Original Russian text published in Azbelev N.V., Maksimov V.P., Rakhmatullina L.F. Elementy sovremennoi teorii funktsional'no-differentsial'nykh uravnenii. Metody i prilozheniya, Moscow: Institute of Computer Science, 2003, 384 p.
  2. El'sgol'ts L.E., Norkin S.B. Introduction to the theory and application of differential equations with deviating arguments, New York: Academic Press, 1973, XVI+357 p. Original Russian text published in El'sgol'ts L.E., Norkin S.B. Vvedenie v teoriyu differentsial'nykh uravnenii s otklonyayushchimsya argumentom, Moscow: Nauka, 1971, 296 p.
  3. Hale J.K., Lunel S.M.V. Introduction to functional differential equations, New York: Springer, 1993, X+450 p.
  4. Smith H. An introduction to delay differential equations with sciences applications to the life, New York: Springer, 2011, XI+172 p.
  5. Tsar'kov E.F. Sluchainye vozmushcheniya differentsial'no-funktsional'nykh uravnenii (Random perturbations of differential-functional equations), Riga: Zinatne, 1989, 421 p.
  6. Kushner H.J. Numerical methods for controlled stochastic delay systems, Boston: Birkhäuser, 2008, XX+282 p.
  7. Mohammed S.E.A. Stochastic functional differential equations, Boston-London: Pitman Publishing, 1984, IX+245 p.
  8. Gardiner C.W. Handbook of stochastic methods for physics, chemistry and the natural sciences, Berlin: Springer-Verlag, 1985, 442 p. Translated under the title Stokhasticheskie metody v estestvennykh naukakh, Moscow: Mir, 1986, 528 p.
  9. Pugachev V.S., Sinitsyn I.N. Stokhasticheskie differentsial'nye sistemy. Analiz i fil'tratsiya (Stochastic differential systems: analysis and filtration), Moscow: Nauka, 1990, 630 p.
  10. Malanin V.V., Poloskov I.E. Sluchainye protsessy v nelineinykh dinamicheskikh sistemakh. Analiticheskie i chislennye metody issledovaniya (Random processes in nonlinear dynamic systems. Analytical and numerical methods of analysis), Izhevsk: Regular and Chaotic Dynamics, 2001, 160 p.
  11. Klyatskin V.I. Dinamika stokhasticheskikh sistem: Kurs lektsii (Dynamics of stochastic systems: lecture course), Moscow: Fizmatgiz, 2003, 240 p.
  12. Mao X. Stochastic differential equations and applications, Oxford: Woodhead Publishing, 2010, XVIII+422 p.
  13. Bellen A., Zennaro M. Numerical methods for delay differential equations, Oxford: Oxford University Press, 2003, XIV+395 p.
  14. Shampine L.F., Gladwell I., Thompson S. Solving ODEs with Matlab, Cambridge: Cambridge University Press, 2003, 272 p. Translated under the title Reshenie obyknovennykh differentsial'nykh uravnenii s ispol'zovaniem MATLAB, St. Petersburg: Lan', 2009, 304 p.
  15. Kloeden P.E., Platen E. Numerical solution of stochastic differential equations, Berlin: Springer-Verlag, 1995, XXXV+632 p.
  16. Milstein G.N., Tretyakov M.V. Stochastic numerics for mathematical physics, Berlin-Heidelberg: Springer-Verlag, 2004, XIX+594 p.
  17. Solodov A.V., Solodova E.A. Sistemy s peremennym zapazdyvaniem (Systems with variable delay), Moscow: Nauka, 1980, 384 p.
  18. Skanavi G.I. Fizika dielectrikov (oblast' sil'nykh polei) (Physics of dielectrics (stronger fields)), Moscow: GIFML, 1958, 907 p.
  19. Sysoev Yu.A., Plankovskii S.I., Loyan A.V., Koshelev N.N. Excitation in a high arc plasma generator, Aviatsionno-Kosmicheskaya Tekhnika i Tekhnologiya, 2006, no. 10, pp. 61-66 (in Russian).
  20. Prochazka I., Kral L., Blazej J. Picosecond laser pulse distortion by propagation through a turbulent atmosphere, Coherence and Ultrashort Pulse Laser Emission, F.J. Duarte (ed.), Rijeka, Croatia: InTech, 2010, pp. 445-448.
  21. Forde J.E. Delay differential equation models in mathematical biology, PhD thesis, University of Michigan, 2005, 94 p.
  22. Crauel H., Son D.T., Siegmund S. Difference equations with random delay, Journal of Difference Equations and Applications, 2009, vol. 15, no. 7, pp. 627-647.
  23. Lara-Sagahon A.V., Kharchenko V., Jose M.V. Stability analysis of a delay-difference SIS epidemiological model, Applied Mathematical Sciences, 2007, vol. 1, no. 26, pp. 1277-1298.
  24. Cooke K.L., Kuang Y., Li B. Analysis of an antiviral immune response model with time delays, Canadian Appl. Math. Quart., 1998, vol. 6, pp. 321-354.
  25. Poddubnyi V.V., Romanovich O.V. Dynamic market model of Walras type with random delays in the supply of goods, Sovremennye napravleniya teoreticheskikh i prikladnykh nauk'2007: Sbornik nauchnykh trudov po materialam nauchno-prakticheskoi konferentsii (Modern branches of theoretical and applied sciences: Transactions on materials of the international scientific-practical conference), Odessa: Chernomor'e, 2007, vol. 21, Physics and Mathematics, Geography, Geology, pp. 20-26 (in Russian).
  26. Chang H.-J., Dye C.-Y. An inventory model with stock-dependent demand under conditions of permissible delay in payments, Journal of Statistics and Management Systems, 1999, vol. 2, no. 2/3, pp. 117-126.
  27. Shepp L. A model for stock price fluctuations based on information, IEEE Trans. on Information Theory, 2002, vol. 48, no. 6, pp. 1372-1378.
  28. Huang D., Nguang S.K. Robust control for uncertain networked control systems with random delays, London: Springer, 2009, XII+168 p.
  29. Ge Y., Chen Q., Jiang M., Huang Y. Modeling of random delays in networked control systems, Journal of Control Science and Engineering, 2013, vol. 2013, Article ID 383415, 9 p.
  30. Lidskii E.A. On the stability of system motions with random delays, Differ. Uravn., 1965, vol. 1, no. 1, pp. 96-101 (in Russian).
  31. Kats I.Ya. The stability on the first approximation of systems with random delay, Prikl. Mat. Mekh., 1967, vol. 31, no. 3, pp. 447-452 (in Russian).
  32. Kolomiets V.G., Korenevskii D.G. Excitation of oscillations in nonlinear systems with random delay, Ukr. Mat. Zh., 1966, vol. 18, no. 3, pp. 51-57 (in Russian).
  33. Korenevskii D.G., Kolomiets V.G. Some questions in the theory of nonlinear oscillations of quasi-linear systems with random delay, Matematicheskaya Fizika, Kiev, 1967, no. 3, pp. 91-113 (in Russian).
  34. Novakovskaya L.I. Construction of asymptotic solutions for the first order differential equations with random delay, Ukr. Mat. Zh., 1989, vol. 41, no. 11, pp. 1569-1563 (in Russian).
  35. Krapivsky P.L., Luck J.M., Mallick K. On stochastic differential equations with random delay, 2011, arXiv:1108.1298 [cond-mat.stat-mech].
  36. Caraballo T., Kloeden P.E., Real J. Discretization of asymptotically stable stationary solutions of delay differential equations with a random stationary delay, Journal of Dynamics and Differential Equations, 2006, vol. 18, no. 4, pp. 863-880.
  37. Zhang H., Feng G., Han C. Linear estimation for random delay systems, Systems & Control Letters, 2011, vol. 60, no. 7, pp. 450-459.
  38. Gao Sh.-L. Generalized stochastic resonance in a linear fractional system with a random delay, Journal of Statistical Mechanics: Theory and Experiment, 2012, vol. 2012, no. 12, P12011.
  39. Mier-y-Teran-Romero L., Lindley B., Schwartz I.B. Statistical multi-moment bifurcations in random-delay coupled swarms, Phys. Rev. E., 2012, vol. 86, no. 5, 056202 (4 p).
  40. Masoller C., Marti A.C. Random delays and the synchronization of chaotic maps, Phys. Rev. Letters, 2005, vol. 94, no. 13, 134102 (4 p).
  41. Wu F., Yin G., Wang L.Y. Moment exponential stability of random delay systems with two-time-scale Markovian switching, Nonlinear Analysis: Real World Applications, 2012, vol. 13, no. 6, pp. 2476-2490.
  42. Tikhonov V.I., Mironov M.A. Markovskie protsessy (Markov processes), Moscow: Sov. Radio, 1977, 488 p.
  43. Law A.M., Kelton W.D. Simulation modeling and analysis, 3d ed., New York: McGraw-Hill, 2000, 784 p. Translated under the title Imitatsionnoe modelirovanie: Klassika CS, 3-e izd., Saint Petersburg: Piter, 2004, 847 p.
  44. Mitropolskii Yu.A. Metod usredneniya v nelineinoi mekhanike (The method of averaging in nonlinear mechanics), Kiev: Naukova dumka, 1970, 440 p.
  45. Kazmerchuk Y.I., Wu J.H. Stochastic state-dependent delay differential equations with applications in finance, Funct. Diff. Equations, 2004, vol. 11, no. 1/2, pp. 77-86.
  46. Zaitsev V.V., Karlov-junior A.V., Telegin S.S. The discrete time “predator-prey” model, Vestn. Samar. Gos. Univ., Estestvennonauchn. Ser., 2009, no. 6 (72), pp. 139-148 (in Russian).
  47. Poloskov I.E. Phase space extension in the analysis of differential-difference systems with random input, Automation and Remote Control, 2002, vol. 63, no. 9, pp. 1426-1438.
  48. Poloskov I.E. Symbolic-numeric algorithms for analysis of stochastic systems with different forms of aftereffect, Proc. in Applied Math. and Mechanics (PAMM), 2007, vol. 7, no. 1, pp. 2080011-2080012.
  49. Poloskov I.E. Symbolic and numeric schemes of analysis of dynamic systems with aftereffect, Vestn. Perm. Univ. Mat. Mekh. Informatika, 2011, no. 2 (6), pp. 51-58 (in Russian).
  50. Mangano S. Mathematica cookbook, Cambridge: O'Reilly Media, Inc., 2010, XXIV+800 p.
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