Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T11:54:57.967Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic continuous-time model reference adaptive systems with decreasing gain

Published online by Cambridge University Press:  01 July 2016

Knut Kristian Aase
Knut Kristian Aase*
Affiliation:
Telemark Distriktshøgskole
*
∗Present address: Norwegian School of Economics and Business Adminstration, 5000 Bergen, Norway.
Get access Rights & Permissions [Opens in a new window]

Abstract

Recursive estimation is considered for parameters of certain continuous stochastic models. Several optimality properties are shown to hold for the resulting recursive estimator, where a stochastic approximation viewpoint is taken when deriving statistical properties, like strong consistency and convergence in distribution. Applications are considered throughout, where for example explosion theory for diffusion processes is used as a modeling guide, in a particular application.

Keywords

STOCHASTIC DIFFERENTIAL EQUATIONSITO PROCESSESRECURSIVE ESTIMATIONEXPLOSION FOR DIFFUSIONSORNSTEIN–UHLENBECK PROCESSLOGISTIC MODELS
Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 4 , December 1982 , pp. 763 - 788
Copyright
Copyright © Applied Probability Trust 1982 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research partly carried out while the author was visiting the University of California, Berkeley.

References

Aase, K. K. (1976) A note on a singular diffusion equation in population genetics. J. Appl. Prob. 13, 1–8.Google Scholar
Aase, K. K. (1977) Conditioned moments of time to fixation. J. Math. Biol. 4, 323–326.CrossRefGoogle Scholar
Aase, K. K. (1977b) A conditional expectation formula for diffusion processes. J. Appl. Prob. 14, 626–629.Google Scholar
Aase, K. K. (1981) Model reference adaptive systems applied to regression theory. Statist. Neerlandica 3, 129–155.Google Scholar
Aase, K. K. (1982) Recursive estimation in nonlinear time series models of autoregressive type. J. R. Statist. Soc. B 44.Google Scholar
Breiman, L. (1968) Probability. Addision-Wesley, Reading, Ma.Google Scholar
Brillinger, D. R. (1975) Time Series. Data Analysis and Theory. Holt, Rinehart and Winston, New York.Google Scholar
Brillinger, D. R. (1981) Some aspects of modern population mathematics. Canad. J. Statist. 9, 173–194.CrossRefGoogle ScholarPubMed
Brown, B. M. and Hewitt, J. I. (1975) Asymptotic likelihood theory for diffusion processes. J. Appl. Prob. 12, 228–238.Google Scholar
Doob, J. L. (1937) Stochastic processes depending on a continuous parameter. Trans. Amer. Math. Soc. 42, 107–140.Google Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.Google Scholar
Fristedt, B. and Orey, S. (1978) The tail s-field of one-dimensional diffusions. In Stochastic Analysis, ed. Friedman, A. and Pinsky, M., Academic Press, New York, 127–138.Google Scholar
Kalman, R. E. (1960) A new approach to linear filtering and prediction problems. J. Basic Engineering D 82, 35–45.Google Scholar
Kushner, T. (1980) Some alternatives in recursive estimation. Internat. J. Control 32, 311–328.Google Scholar
Kushner, H. J. (1977) Convergence of recursive adaptive and identification procedures via weak convergence theory. IEEE Trans. Automatic Control AC-22, 921–930.Google Scholar
Landau, I. D. (1976) Unbiased recursive identification using model reference adaptive techniques. IEEE Trans. Automatic Control AC-21, 194–202.Google Scholar
Li, T. Y. and Yorke, J. A. (1975) Period three implies chaos. Amer. Math. Monthly 82, 985–992.Google Scholar
Lipster, R. S. and Shiryayev, A. N. (1978) Statistics of Random Processses, Vols. 1 and 2. Springer-Verlag, Berlin.Google Scholar
Ljung, L. (1977) On positive real transfer functions and the convergence of some recursive schemes. IEEE Trans. Automatic Control AC-22, 539–551.CrossRefGoogle Scholar
McKean, H. P. Jr. (1969) Stochastic Integrals. Academic Press, New York.Google Scholar
Mayer-Kress, G. and Haken, H. (1981) The influence of noise on the logistic model. J. Statist. Phys. 26, 149–171.Google Scholar
Merton, R. C. (1971) Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3, 373–413.Google Scholar
Sussman, H. J. (1978) On the gap between deterministic and stochastic ordinary differential equations. Ann. Prob. 6, 19–41.Google Scholar
Taraskin, A. F. (1974) On the asymptotic normality of vector-valued stochastic integrals and estimates of drift parameters of a multidimensional diffusion process. Theory Prob. Math. Statist. 2, 209–224.Google Scholar
Tucker, H. G. (1967) A Graduate Course in Probability. Academic Press, New York.Google Scholar
Wong, E. (1971) Stochastic Processes in Information and Dynamical Systems. McGraw-Hill, New York.Google Scholar