Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-16T04:23:34.042Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for stochastic difference-differential equations

Published online by Cambridge University Press:  22 January 2016

Tsukasa Fujiwara  and
Hiroshi Kunita
Tsukasa Fujiwara*
Affiliation:
Department of Applied Science, Kyushu University, 36 Fukuoka 812, Japan
Hiroshi Kunita*
Affiliation:
Department of Applied Science, Kyushu University, 36 Fukuoka 812, Japan
*
Department of Mathematics, Hyogo University of Teacher, Education Yashiro, Hyogo 673-14, Japan
Department of Mathematics, Hyogo University of Teacher, Education Yashiro, Hyogo 673-14, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

There are extensive works on the limit theorems for sequences of stochastic ordinary differential equations written in the form:

where is a stochastic process and is a deterministic function, both of which take values in the space of vector fields. The case where {ftn} n satisfies certain mixing conditions has been studied by Khas’minskii [7], Kesten-Papanicolaou [6] and others.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 127 , September 1992 , pp. 83 - 116
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1992

References

[ 1 ] Aldous, D., Stopping times and tightness, Ann. Probab., 6 (1978), 335–340.Google Scholar
[ 2 ] Fujiwara, T., On the jump-diffusion approximation of stochastic difference equations driven by a mixing sequence, J. Math. Soc. Japan, 42 (1990), 353–376.Google Scholar
[ 3 ] Fujiwara, T., Limit theorems for random difference equations driven by mixing processes, to appear in J. Math. Kyoto Univ.Google Scholar
[ 4 ] Ikeda, N. and Watanabe, S., Stochastic Differential Equations and Diffusion Processes, North-Holland/Kodansha (1981).Google Scholar
[ 5 ] Jacod, J. and Shiryaev, A. N., Limit Theorems for Stochastic Processes, Springer-Verlag (1987).CrossRefGoogle Scholar
[ 6 ] Kesten, H. and Papanicolaou, G.C., A limit theorem for turbulent diffusion, Comm. Math. Phys., 65 (1979), 97–128.Google Scholar
[ 7 ] Khas’minskii, R.Z., A limit theorem for the solution of differential equations with random right-hand sides, Theory Probab. Appl., 11 (1966), 390–406.Google Scholar
[ 8 ] Kunita, H., Stochastic Flows and Applications, Tata Institute of Fundamental Research, Springer, (1986).Google Scholar
[ 9 ] Kunita, H., Stochastic Flows and Stochastic Differential Equations, Cambridge Univ. Press, (1990).Google Scholar
[10] Kunita, H., Central limit theorems on random measures and stochastic difference equations, Gaussian Random Fields, ed. Itô, K. & Hida, T., World Scientific, (1991), 28–42.Google Scholar
[11] Kunita, H., Limits on random measures and stochastic difference equations related to mixing array of random variables, Stochastic Analysis, ed. Barlow, M.T. & Bingham, N.H., LMS Lecture Note Series 167, Cambridge Univ. Press, (1991), 229–254.Google Scholar
[12] Kurtz, T.G., Approximation of Population Processes, Reginal Conference Series in Applied Mathematics 36, SIAM, (1981).Google Scholar
[13] Kushner, H. J., Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic System Theory, MIT Press, (1984).Google Scholar
[14] Meyer, P.A., Probabilités et Potentiel, Herman, (1966).Google Scholar
[15] Słomiński, L., Stability of strong solutions of stochastic differential equations, Stochastic Process. Appl., 31 (1989), 173–202.Google Scholar
[16] Stroock, D.W. and Varadhan, S.R.S., Multidimensional Diffusion Processes, Springer-Verlag, (1979).Google Scholar
[17] Watanabe, H., Diffusion approximations of stochastic difference equations II, Hiroshima Math. J., 14 (1984), 15–34.Google Scholar
[18] Wong, E. and Zakai, M., On the relation between ordinary and stochastic differential equations, Int. J. Eng. Sci., 3 (1965), 213–229.Google Scholar