On Diffusion Approximation of Discrete Markov Dynamical Systems

The paper is devoted to stochastic analysis of finite dimensional difference equation with dependent on ergodic Markov chain increments, which are proportional to small parameter ". A point-form solution of this difference equation may be represented as vertexes of a time-dependent continuous broken line given on the segment [0,1] with "-dependent scaling of intervals between vertexes. Tending " to zero one may apply stochastic averaging and diffusion approximation procedures and construct continuous approximation of the initial stochastic iterations as an ordinary or stochastic Ito differential equation. The paper proves that for sufficiently small " these equations may be successfully applied not only to approximate finite number of iterations but also for asymptotic analysis of iterations, when number of iterations tends to infinity.

Authors:

• Jevgenijs Carkovs

Keywords:

• Markov dynamical system
• diffusion approximation
• equilibrium stochastic stability.

References:

[1] C.M. Ann and H. Thompson, Jump-diffusion process and term structure
of interest rates.J. of Finance, 43, No. 3, 155-174. 1988, 155-174.
[2] L. Arnold, G. Papanicolaou, and V. Wihstutz, Asymptotic analysis of the
Lyapunov exponent and rotation number of the random oscillator and
applications, SIAM J. Appl. Math. 46, No. 3, , 1986, 427-450.
[3] C.A. Ball and A.C. Roma, A jump diffusion model for the European
Monetary System. J. of International Money and Finance, 12 1993,
475-492..
[4] G. Blankenship, and G. Papanicolaou, Stability and control of stochastic
systems with wide-band noise disturbances.I. SIAM J. Appl. Math. 34,
No. 3, 1978, 437-476.
[5] M. Dimentberg, Statistical Dynamics of Nonlinear and Time-Varying
Systems. NY, USA: Willey, 1988.
[6] E.B. Dynkin, Markov Processes, NY, USA: Academic Press, 1965.
[7] F. Fornari and A. Mele, Sign- and volatility-switching ARCH models
theory and applications to international stock markets. J. of Applied
Econometrics, 12, 1997, 49-65.
[8] R. Jarrow and E. Rosenfeld, Jump risks and the International capital asset
pricing models. J. of Business, 57, 1984, 337-351.
[9] M. Jeanblanc-Pickue M. Pontier, Optimal portfolio for a small investor in
a market model with discontinuous prices. Appl. Math. Optim., 22,1990
287-310.
[10] R.Z. Khasminsky, Limit theorem for a solution of the differential
equation with a random right part. Prob. Theor. and its Appl., 11, No 3,
1966, 444 - 462.
[11] R.Z. Khasminsky, Stochastic Stability of Differential Equations, MA,
USA: Kluver Academic Pubs. 1980.
[12] D.B. Nelson, ARCH models as diffusion approximation.J. of Econometrics,
45, 1990, 7-38.
[13] M.B. Nevelson and R.Z. Hasminskii. Stochastic Approximation and
Recursive Estimation, NY, USA:American Mathematical Society, ISBN-
10:0821809067, 1976
[14] A.V. Skorokhod, Studies in the theory of random processes, 3rd ed.
Reading, USA: Addison-Wesley publishing, 1965.
[15] A. V. Skorokhod, Asymptotic Methods of Theory of Stochastic Differential
Equations, 3rd ed. AMS, USA: Providance, 1994.
[16] Ye. Tsarkov (J.Carkovs), Averaging in Dynamical Systems with Markov
Jumps, Preprint No. 282, April, Institute of Dynamical Systems Bremen
University, Germany, 1993.
[17] Ye. Tsarkov (J.Carkovs), Asymptotic methods for stability analysis of
Markov impulse dynamical systems, Nonlinear dynamics and system
theory, 1, No. 2 , 2002,103 - 115.
[18] E. Wong.The construction of a class of stationary Markov process.
(R.Bellman ed.), Proceedings of the 16th Symposia in Applied Mathematics:
Stochastc Processes in Mathematical Physics and Engineering
Providance, RI, 1964, 264-276.