# On the infinite time horizon linear-quadratic regulator problem under a fractional Brownian perturbation

Marina L. Kleptsyna; Alain Le Breton; Michel Viot

ESAIM: Probability and Statistics (2010)

- Volume: 9, page 185-205
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topKleptsyna, Marina L., Le Breton, Alain, and Viot, Michel. "On the infinite time horizon linear-quadratic regulator problem under a fractional Brownian perturbation." ESAIM: Probability and Statistics 9 (2010): 185-205. <http://eudml.org/doc/104329>.

@article{Kleptsyna2010,

abstract = {
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.
},

author = {Kleptsyna, Marina L., Le Breton, Alain, Viot, Michel},

journal = {ESAIM: Probability and Statistics},

keywords = {Fractional Brownian motion; linear system;
optimal control; quadratic payoff; infinite time.; optimal control; infinite time},

language = {eng},

month = {3},

pages = {185-205},

publisher = {EDP Sciences},

title = {On the infinite time horizon linear-quadratic regulator problem under a fractional Brownian perturbation},

url = {http://eudml.org/doc/104329},

volume = {9},

year = {2010},

}

TY - JOUR

AU - Kleptsyna, Marina L.

AU - Le Breton, Alain

AU - Viot, Michel

TI - On the infinite time horizon linear-quadratic regulator problem under a fractional Brownian perturbation

JO - ESAIM: Probability and Statistics

DA - 2010/3//

PB - EDP Sciences

VL - 9

SP - 185

EP - 205

AB -
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.

LA - eng

KW - Fractional Brownian motion; linear system;
optimal control; quadratic payoff; infinite time.; optimal control; infinite time

UR - http://eudml.org/doc/104329

ER -

## References

top- 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. Zbl1064.93048
- D. Blackwell and L. Dubins, Merging of opinions with increasing information. Ann. Math. Statist.33 (1962) 882–886. Zbl0109.35704
- M.H.A. Davis, Linear Estimation and Stochastic Control. Chapman and Hall, New York (1977). Zbl0437.60001
- L. Decreusefond and A.S. Üstünel, Stochastic analysis of the fractional Brownian motion. Potential Anal.10 (1999) 177–214. Zbl0924.60034
- 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. Zbl0947.60061
- G. Gripenberg and I. Norros, On the prediction of fractional Brownian motion. J. Appl. Probab.33 (1996) 400–410. Zbl0861.60049
- M.L. Kleptsyna and A. Le Breton, Statistical analysis of the fractional Ornstein-Uhlenbeck type process. Statist. Inference Stochastic Processes5 (2002) 229–248. Zbl1021.62061
- M.L. Kleptsyna and A. Le Breton, Extension of the Kalman-Bucy filter to elementary linear systems with fractional Brownian noises. Statist. Inference Stochastic Processes5 (2002) 249–271. Zbl1011.60018
- M.L. Kleptsyna, A. Le Breton and M.-C. Roubaud, General approach to filtering with fractional Brownian noises – Application to linear systems. Stochastics Reports71 (2000) 119–140. Zbl0979.93117
- M.L. Kleptsyna, A. Le Breton and M. Viot, About the linear-quadratic regulator problem under a fractional Brownian perturbation. ESAIM: PS7 (2003) 161–170. Zbl1030.93059
- 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. Zbl1274.60117
- A. Le Breton, Adaptive control in the scalar linear-quadratic model in continious time. Statist. Probab. Lett.13 (1992) 169–177. Zbl0744.93090
- 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). Zbl0482.60030
- G.M. Molchan, Linear problems for fractional Brownian motion: group approach. Probab. Theory Appl.1 (2002) 59–70 (in Russian). Zbl1035.60084
- G.M. Molchan, Gaussian processes with spectra which are asymptotically equivalent to a power of λ. Probab. Theory Appl.14 (1969) 530–532.
- G.M. Molchan and J.I. Golosov, Gaussian stationary processes with which are asymptotic power spectrum. Soviet Math. Dokl.10 (1969) 134–137. Zbl0181.20704
- I. Norros, E. Valkeila and J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli5 (1999) 571–587. Zbl0955.60034
- C.J. Nuzman and H.V. Poor, Linear estimation of self-similar processes via Lamperti's transformation. J. Appl. Prob.37 (2000) 429–452. Zbl0963.60034

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.