Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T22:58:38.932Z Has data issue: false hasContentIssue false
  • English
  • Français

On the infinite time horizon linear-quadratic regulator problem undera fractional Brownian perturbation

Published online by Cambridge University Press:  15 November 2005

Marina L. Kleptsyna ,
Alain Le Breton  and
Michel Viot
Marina L. Kleptsyna
Affiliation:
Laboratoire de Statistique et Processus, Université du Maine, av. Olivier Messiaen, 72085 Le Mans Cedex 9, France; marina.kleptsyna@univ-lemans.fr
Alain Le Breton
Affiliation:
Laboratoire de Modélisation et Calcul, Université J. Fourier, BP 53, 38041 Grenoble Cedex 9, France; Alain.Le-Breton@imag.fr
Michel Viot
Affiliation:
Laboratoire de Modélisation et Calcul, Université J. Fourier, BP 53, 38041 Grenoble Cedex 9, France; Alain.Le-Breton@imag.fr
Get access

Abstract

In this paper we solve the basic fractionalanalogue of the classical infinite time horizon linear-quadratic Gaussianregulator problem. For a completely observable controlled linearsystem driven by a fractional Brownian motion, we describeexplicitely the optimal control policy which minimizes anasymptotic quadratic performance criterion.

Keywords

Fractional Brownian motionlinear systemoptimal controlquadratic payoffinfinite time.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 9 , February 2005 , pp. 185 - 205
Copyright
© EDP Sciences, SMAI, 2005

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.)

References

Biaggini, F., Hu, Y., Øksendal, B. and Sulem, A., A stochastic maximum principle for processes driven by fractional Brownian motion. Stochastic Processes Appl. 100 (2002) 233–253.
Blackwell, D. and Dubins, L., Merging of opinions with increasing information. Ann. Math. Statist. 33 (1962) 882–886. CrossRef
M.H.A. Davis, Linear Estimation and Stochastic Control. Chapman and Hall, New York (1977).
Decreusefond, L. and Üstünel, A.S., Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1999) 177–214. CrossRef
Duncan, T.E., Hu, Y. and Pasik-Duncan, B., Stochastic calculus for fractional Brownian motion I. Theory. SIAM J. Control Optim. 38 (2000) 582–612. CrossRef
Gripenberg, G. and Norros, I., On the prediction of fractional Brownian motion. J. Appl. Probab. 33 (1996) 400–410. CrossRef
Kleptsyna, M.L. and Le Breton, A., Statistical analysis of the fractional Ornstein-Uhlenbeck type process. Statist. Inference Stochastic Processes 5 (2002) 229–248. CrossRef
Kleptsyna, M.L. and Le Breton, A., Extension of the Kalman-Bucy filter to elementary linear systems with fractional Brownian noises. Statist. Inference Stochastic Processes 5 (2002) 249–271. CrossRef
Kleptsyna, M.L., Le Breton, A. and Roubaud, M.-C., General approach to filtering with fractional Brownian noises – Application to linear systems. Stochastics Reports 71 (2000) 119–140.
Kleptsyna, M.L., Le Breton, A. and Viot, M., About the linear-quadratic regulator problem under a fractional Brownian perturbation. ESAIM: PS 7 (2003) 161–170. CrossRef
M.L. Kleptsyna, A. Le Breton and M. Viot, Asymptotically optimal filtering in linear systems with fractional Brownian noises. Statist. Oper. Res. Trans. (2004) 28 177–190.
Le Breton, A., Adaptive control in the scalar linear-quadratic model in continious time. Statist. Probab. Lett. 13 (1992) 169–177. CrossRef
R.S. Liptser and A.N. Shiryaev, Statist. Random Processes. Springer-Verlag, New York (1978).
R.S. Liptser and A.N. Shiryaev, Theory of Martingales. Kluwer Academic Publ., Dordrecht (1989).
Molchan, G.M., Linear problems for fractional Brownian motion: group approach. Probab. Theory Appl. 1 (2002) 59–70 (in Russian).
Molchan, G.M., Gaussian processes with spectra which are asymptotically equivalent to a power of λ. Probab. Theory Appl. 14 (1969) 530–532.
Molchan, G.M. and Golosov, J.I., Gaussian stationary processes with which are asymptotic power spectrum. Soviet Math. Dokl. 10 (1969) 134–137.
Norros, I., Valkeila, E. and Virtamo, J., An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5 (1999) 571–587. CrossRef
Nuzman, C.J. and Poor, H.V., Linear estimation of self-similar processes via Lamperti's transformation. J. Appl. Prob. 37 (2000) 429–452. CrossRef