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 : 10.1051/ps:2005008
Classification : 60G15, 60G44, 93E20
Mots-clés : fractional brownian motion, linear system, optimal control, quadratic payoff, infinite time 
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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. https://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

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

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

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

[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

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

[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

[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

[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

[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 | Numdam | Zbl

[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 | Zbl

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

[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 | Zbl

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

[16] G.M. Molchan, Gaussian processes with spectra which are asymptotically equivalent to a power of λ. 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

[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

[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

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