On the infinite time horizon linear-quadratic regulator problem under a fractional brownian perturbation
ESAIM: Probability and Statistics, Tome 9 (2005), pp. 185-205.

In this paper we solve the basic fractional analogue of the classical infinite time horizon linear-quadratic gaussian regulator problem. For a completely observable controlled linear system driven by a fractional brownian motion, we describe explicitely the optimal control policy which minimizes an asymptotic quadratic performance criterion.

DOI : https://doi.org/10.1051/ps:2005008
Classification : 60G15,  60G44,  93E20
Mots clés : fractional brownian motion, linear system, optimal control, quadratic payoff, infinite time
@article{PS_2005__9__185_0,
author = {Kleptsyna, Marina L. and Breton, Alain Le and Viot, Michel},
title = {On the infinite time horizon linear-quadratic regulator problem under a fractional brownian perturbation},
journal = {ESAIM: Probability and Statistics},
pages = {185--205},
publisher = {EDP-Sciences},
volume = {9},
year = {2005},
doi = {10.1051/ps:2005008},
zbl = {1136.93463},
mrnumber = {2148966},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2005008/}
}
TY  - JOUR
AU  - Kleptsyna, Marina L.
AU  - Breton, Alain Le
AU  - Viot, Michel
TI  - On the infinite time horizon linear-quadratic regulator problem under a fractional brownian perturbation
JO  - ESAIM: Probability and Statistics
PY  - 2005
DA  - 2005///
SP  - 185
EP  - 205
VL  - 9
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2005008/
UR  - https://zbmath.org/?q=an%3A1136.93463
UR  - https://www.ams.org/mathscinet-getitem?mr=2148966
UR  - https://doi.org/10.1051/ps:2005008
DO  - 10.1051/ps:2005008
LA  - en
ID  - PS_2005__9__185_0
ER  - 
Kleptsyna, Marina L.; Breton, Alain Le; Viot, Michel. On the infinite time horizon linear-quadratic regulator problem under a fractional brownian perturbation. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 185-205. doi : 10.1051/ps:2005008. http://www.numdam.org/articles/10.1051/ps:2005008/

[1] F. Biaggini, Y. Hu, B. Øksendal and A. Sulem, A stochastic maximum principle for processes driven by fractional Brownian motion. Stochastic Processes Appl. 100 (2002) 233-253. | Zbl 1064.93048

[2] D. Blackwell and L. Dubins, Merging of opinions with increasing information. Ann. Math. Statist. 33 (1962) 882-886. | Zbl 0109.35704

[3] M.H.A. Davis, Linear Estimation and Stochastic Control. Chapman and Hall, New York (1977). | MR 476099 | Zbl 0437.60001

[4] L. Decreusefond and A.S. Üstünel, Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1999) 177-214. | Zbl 0924.60034

[5] T.E. Duncan, Y. Hu and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion I. Theory. SIAM J. Control Optim. 38 (2000) 582-612. | Zbl 0947.60061

[6] G. Gripenberg and I. Norros, On the prediction of fractional Brownian motion. J. Appl. Probab. 33 (1996) 400-410. | Zbl 0861.60049

[7] M.L. Kleptsyna and A. Le Breton, Statistical analysis of the fractional Ornstein-Uhlenbeck type process. Statist. Inference Stochastic Processes 5 (2002) 229-248. | Zbl 1021.62061

[8] M.L. Kleptsyna and A. Le Breton, Extension of the Kalman-Bucy filter to elementary linear systems with fractional Brownian noises. Statist. Inference Stochastic Processes 5 (2002) 249-271. | Zbl 1011.60018

[9] M.L. Kleptsyna, A. Le Breton and M.-C. Roubaud, General approach to filtering with fractional Brownian noises - Application to linear systems. Stochastics Reports 71 (2000) 119-140. | Zbl 0979.93117

[10] M.L. Kleptsyna, A. Le Breton and M. Viot, About the linear-quadratic regulator problem under a fractional Brownian perturbation. ESAIM: PS 7 (2003) 161-170. | EuDML 245312 | Numdam | Zbl 1030.93059

[11] 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. | EuDML 40457 | Zbl 1274.60117

[12] A. Le Breton, Adaptive control in the scalar linear-quadratic model in continious time. Statist. Probab. Lett. 13 (1992) 169-177. | Zbl 0744.93090

[13] R.S. Liptser and A.N. Shiryaev, Statist. Random Processes. Springer-Verlag, New York (1978).

[14] R.S. Liptser and A.N. Shiryaev, Theory of Martingales. Kluwer Academic Publ., Dordrecht (1989). | MR 1022664 | Zbl 0728.60048

[15] G.M. Molchan, Linear problems for fractional Brownian motion: group approach. Probab. Theory Appl. 1 (2002) 59-70 (in Russian). | Zbl 1035.60084

[16] G.M. Molchan, Gaussian processes with spectra which are asymptotically equivalent to a power of $\lambda$. Probab. Theory Appl. 14 (1969) 530-532.

[17] G.M. Molchan and J.I. Golosov, Gaussian stationary processes with which are asymptotic power spectrum. Soviet Math. Dokl. 10 (1969) 134-137. | Zbl 0181.20704

[18] I. Norros, E. Valkeila and J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5 (1999) 571-587. | Zbl 0955.60034

[19] C.J. Nuzman and H.V. Poor, Linear estimation of self-similar processes via Lamperti's transformation. J. Appl. Prob. 37 (2000) 429-452. | Zbl 0963.60034

Cité par Sources :