Abstract
The sample path rate of convergence is obtained for a strongly consistent, recursive estimator of a parameter in a bilinear stochastic differential equation. The bilinear stochastic differential equation arises in a model of portfolio selection and consumption. The parameters of this equation change with time and converge to limits. It is assumed that one of these parameters is unknown. In this case it is necessary simultaneously to estimate the unknown parameter and to control the state equation so there is the problem of adaptive control. Asymptotic normality for a parameter estimator is also given.
This work was supported by NSF Grant ECS-8718026.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
Davis, M. H. A., Local Time on the Stock Exchange, Stochastic Calculus in Application, ed. J. R. Norris, Pitman Notes in Mathematics, Longman, London 1988, to appear.
Duncan, T. E. and Pasik-Duncan, B., Adaptive Control of Continuous Time Portfolio and Consumption Model, Journal of Optimization Theory and Applications, Vol. 61, April, 1989.
Duncan, T. E. and Pasik-Duncan, B., Adaptive Control of Three Continuous Time Portfolio and Consumption Models, Journal of Optimization Theory and Applications, Vol. 61, June 1989.
Duncan, T. E. and Pasik-Duncan, B., The Rate of Convergence for an Estimator in a Portfolio and Consumption Model, Journal of Optimization Theory and Applications, Vol. 61, May 1989.
Karatzas, I., Lahoczky, J. P. and Shreve, S. E., Optimal portfolio and consumption decision for a "small investor" on a finite horizon, SIAM J. Control & Optimization, 25, 1557–1586, 1987.
Kumar, P. R., Adaptive Control with a Compact Parameter Set, SIAM Journal on Control and Optimization, Vol. 20, p. 9–13, 1982.
Kushner, H. and Kumar, R., Convergence and Rate of Convergence of a Recursive Identification and Adaptive Control Model Which Uses Truncated Estimates, IEEE Transactions on Automatic Control, Vol. AC-27, pp. 775–782, 1982.
Mandl. P. and Hübner, G., Transient Phenomena and Self-Optimizing Control of Markov Chains, Acta Universitatis Carolinae-Mathematica et Physica, Vol. 26, No. 1, pp. 35–51, 1985.
Merton, R. C., Optimum Consumption and Portfolio Rules in Continuous-Time Model, Journal of Economic Theory, Vol. 3, pp. 373–413, 1971.
Fleming, W. H. and Rishel, R. W., Deterministic and Stochastic Optimal Control, Springer, Berlin, Germany 1975.
McKean, H. P., Jr., Stochastic Integrals, Academic Press, New York, 1960.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag
About this paper
Cite this paper
Pasik-Duncan, B. (1989). The rate of convergence and the asymptotic normality of an estimator in a controlled investment model with time-varying parameters. In: Christopeit, N., Helmes, K., Kohlmann, M. (eds) Stochastic Differential Systems. Lecture Notes in Control and Information Sciences, vol 126. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0043791
Download citation
DOI: https://doi.org/10.1007/BFb0043791
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51299-8
Online ISBN: 978-3-540-46188-3
eBook Packages: Springer Book Archive