{"title":"On Diffusion Approximation of Discrete Markov Dynamical Systems","authors":"Jevgenijs Carkovs","volume":16,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":206,"pagesEnd":211,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/4982","abstract":"The paper is devoted to stochastic analysis of finite\r\ndimensional difference equation with dependent on ergodic Markov\r\nchain increments, which are proportional to small parameter \". A\r\npoint-form solution of this difference equation may be represented\r\nas vertexes of a time-dependent continuous broken line given on the\r\nsegment [0,1] with \"-dependent scaling of intervals between vertexes.\r\nTending \" to zero one may apply stochastic averaging and diffusion\r\napproximation procedures and construct continuous approximation of\r\nthe initial stochastic iterations as an ordinary or stochastic Ito differential\r\nequation. The paper proves that for sufficiently small \" these\r\nequations may be successfully applied not only to approximate finite\r\nnumber of iterations but also for asymptotic analysis of iterations,\r\nwhen number of iterations tends to infinity.","references":"[1] C.M. Ann and H. Thompson, Jump-diffusion process and term structure\r\nof interest rates.J. of Finance, 43, No. 3, 155-174. 1988, 155-174.\r\n[2] L. Arnold, G. Papanicolaou, and V. Wihstutz, Asymptotic analysis of the\r\nLyapunov exponent and rotation number of the random oscillator and\r\napplications, SIAM J. Appl. Math. 46, No. 3, , 1986, 427-450.\r\n[3] C.A. Ball and A.C. Roma, A jump diffusion model for the European\r\nMonetary System. J. of International Money and Finance, 12 1993,\r\n475-492..\r\n[4] G. Blankenship, and G. Papanicolaou, Stability and control of stochastic\r\nsystems with wide-band noise disturbances.I. SIAM J. Appl. Math. 34,\r\nNo. 3, 1978, 437-476.\r\n[5] M. Dimentberg, Statistical Dynamics of Nonlinear and Time-Varying\r\nSystems. NY, USA: Willey, 1988.\r\n[6] E.B. Dynkin, Markov Processes, NY, USA: Academic Press, 1965.\r\n[7] F. Fornari and A. Mele, Sign- and volatility-switching ARCH models\r\ntheory and applications to international stock markets. J. of Applied\r\nEconometrics, 12, 1997, 49-65.\r\n[8] R. Jarrow and E. Rosenfeld, Jump risks and the International capital asset\r\npricing models. J. of Business, 57, 1984, 337-351.\r\n[9] M. Jeanblanc-Pickue M. Pontier, Optimal portfolio for a small investor in\r\na market model with discontinuous prices. Appl. Math. Optim., 22,1990\r\n287-310.\r\n[10] R.Z. Khasminsky, Limit theorem for a solution of the differential\r\nequation with a random right part. Prob. Theor. and its Appl., 11, No 3,\r\n1966, 444 - 462.\r\n[11] R.Z. Khasminsky, Stochastic Stability of Differential Equations, MA,\r\nUSA: Kluver Academic Pubs. 1980.\r\n[12] D.B. Nelson, ARCH models as diffusion approximation.J. of Econometrics,\r\n45, 1990, 7-38.\r\n[13] M.B. Nevelson and R.Z. Hasminskii. Stochastic Approximation and\r\nRecursive Estimation, NY, USA:American Mathematical Society, ISBN-\r\n10:0821809067, 1976\r\n[14] A.V. Skorokhod, Studies in the theory of random processes, 3rd ed.\r\nReading, USA: Addison-Wesley publishing, 1965.\r\n[15] A. V. Skorokhod, Asymptotic Methods of Theory of Stochastic Differential\r\nEquations, 3rd ed. AMS, USA: Providance, 1994.\r\n[16] Ye. Tsarkov (J.Carkovs), Averaging in Dynamical Systems with Markov\r\nJumps, Preprint No. 282, April, Institute of Dynamical Systems Bremen\r\nUniversity, Germany, 1993.\r\n[17] Ye. Tsarkov (J.Carkovs), Asymptotic methods for stability analysis of\r\nMarkov impulse dynamical systems, Nonlinear dynamics and system\r\ntheory, 1, No. 2 , 2002,103 - 115.\r\n[18] E. Wong.The construction of a class of stationary Markov process.\r\n(R.Bellman ed.), Proceedings of the 16th Symposia in Applied Mathematics:\r\nStochastc Processes in Mathematical Physics and Engineering\r\nProvidance, RI, 1964, 264-276.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 16, 2008"}