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Separation principle in the fractional Gaussianlinear-quadratic regulator problem withpartial observation

Published online by Cambridge University Press:  23 January 2008

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; [email protected]
Alain Le Breton
Affiliation:
Laboratoire de Modélisation et Calcul, Université J. Fourier, BP 53, 38041 Grenoble cedex 9, France; [email protected]
Michel Viot
Affiliation:
Laboratoire de Modélisation et Calcul, Université J. Fourier, BP 53, 38041 Grenoble cedex 9, France; [email protected]
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Abstract

In this paper we solve the basic fractional analogue of the classical linear-quadratic Gaussian regulator problem in continuous-time with partial observation. For a controlled linear system where both the state and observation processes are driven by fractional Brownian motions, we describe explicitly the optimal control policy which minimizes a quadratic performance criterion. Actually, we show that a separation principle holds, i.e., the optimal control separates into two stages based on optimal filtering of the unobservable state and optimal control of the filtered state. Both finite and infinite time horizon problems are investigated.

Keywords

Fractional Brownian motionlinear systemoptimal controloptimal filteringquadratic payoffseparation principle
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 12 , 2008 , pp. 94 - 126
Copyright
© EDP Sciences, SMAI, 2008

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References

Biagini, F., Hu, Y., Øksendal, B., and Sulem, A., A stochastic maximum principle for processes driven by fractional Brownian motion. Stoch. Processes Appl. 100 (2002) 233–253. CrossRef
H. Cramer and M.R. Leadbetter, Stationary and related stochastic processes. John Wiley & Sons, Inc. (1967).
M.H.A. Davis, Linear Estimation and Stochastic Control. Chapman and Hall (1977).
Decreusefond, L. and Üstünel, A.S., Stochastic analysis of the fractional Brownian motion. Potential Analysis 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. Prob. 33 (1996) 400–410. CrossRef
Kleptsyna, M.L. and Le Breton, A., Statistical analysis of the fractional Ornstein-Uhlenbeck type process. Stat. Inf. Stoch. 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. Stat. Inf. Stoch. 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. Stoch. Stoch. 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
Kleptsyna, M.L., Le Breton, A. and Viot, M., Asymptotically optimal filtering in linear systems with fractional Brownian noises. Stat. Oper. Res. Trans. 28 (2004) 177–190.
Kleptsyna, M.L., Le Breton, A. and Viot, M., On the infinite time horizon linear-quadratic regulator problem under a fractional Brownian perturbation. ESAIM: PS 9 (2005) 185–205. CrossRef
R.S. Liptser and A.N. Shiryaev, Statistics of Random Processes. Springer-Verlag (1978).
R.S. Liptser and A.N. Shiryaev, Theory of Martingales. Kluwer Academic Publ., Dordrecht (1989).
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
Wonham, W.M., On the separation principle of stochastic control. SIAM J. Control 6 (1968) 312–326. CrossRef