Separation principle in the fractional gaussian linear-quadratic regulator problem with partial observation
ESAIM: Probability and Statistics, Tome 12 (2008), pp. 94-126.

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.

DOI : 10.1051/ps:2007046
Classification : 93E11, 93E20, 60G15, 60G44
Mots-clés : fractional brownian motion, linear system, optimal control, optimal filtering, quadratic payoff, separation principle 
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     author = {Kleptsyna, Marina L. and Breton, Alain Le and Viot, Michel},
     title = {Separation principle in the fractional gaussian linear-quadratic regulator problem with partial observation},
     journal = {ESAIM: Probability and Statistics},
     pages = {94--126},
     publisher = {EDP-Sciences},
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     year = {2008},
     doi = {10.1051/ps:2007046},
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     zbl = {1136.93463},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2007046/}
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Kleptsyna, Marina L.; Breton, Alain Le; Viot, Michel. Separation principle in the fractional gaussian linear-quadratic regulator problem with partial observation. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 94-126. doi : 10.1051/ps:2007046. http://www.numdam.org/articles/10.1051/ps:2007046/

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

[2] H. Cramer and M.R. Leadbetter, Stationary and related stochastic processes. John Wiley & Sons, Inc. (1967). | MR | Zbl

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

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

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

[7] M.L. Kleptsyna and A. Le Breton, Statistical analysis of the fractional Ornstein-Uhlenbeck type process. Stat. Inf. Stoch. Processes 5 (2002) 229-248. | MR | Zbl

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

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

[11] M.L. Kleptsyna, A. Le Breton and M. Viot, Asymptotically optimal filtering in linear systems with fractional Brownian noises. Stat. Oper. Res. Trans. 28 (2004) 177-190. | MR

[12] M.L. Kleptsyna, A. Le Breton and M. Viot, On the infinite time horizon linear-quadratic regulator problem under a fractional Brownian perturbation. ESAIM: PS 9 (2005) 185-205. | Numdam | MR | Zbl

[13] R.S. Liptser and A.N. Shiryaev, Statistics of Random Processes. Springer-Verlag (1978). | Zbl

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

[15] 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. | MR | Zbl

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

[17] W.M. Wonham, On the separation principle of stochastic control. SIAM J. Control 6 (1968) 312-326. | MR | Zbl

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