# About the linear-quadratic regulator problem under a fractional brownian perturbation

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ps141.dvi ESAIM: Probability and Statistics January 2003, Vol. 7, 161{170 DOI: 10.1051/ps:2003007 ABOUT THE LINEAR-QUADRATIC REGULATOR PROBLEM UNDER A FRACTIONAL BROWNIAN PERTURBATION ∗, ∗∗ M.L. Kleptsyna1, Alain Le Breton2 and M. Viot2 Abstract. In this paper we solve the basic fractional analogue of the classical linear-quadratic Gaussian regulator problem in continuous time. For a completely observable controlled linear system driven by a fractional Brownian motion, we describe explicitely the optimal control policy which minimizes a quadratic performance criterion. Mathematics Subject Classification. 93E20, 60G15, 60G44. Received September 12, 2002. Introduction Several contributions in the literature have been already devoted to the extension of the classical theory of continuous-time stochastic systems driven by Brownian motions to analogues in which the driving processes are fractional Brownian motions (fBm’s for short). The tractability of the standard problems in prediction, parameter estimation and �ltering is now rather well understood (see, e.g. [4,6-8,11,12] and references therein). Concerning optimal control problems, as far as we know, it is far from fully demonstrated (nevertheless, see [5] for an attempt in a general setting and [9] for a study in an elementary archetypal model). Here our aim is to illustrate the actual solvability of control problems by exhibiting an explicit solution for the case of the simplest linear-quadratic model. We deal with the fractional analogue of the so-called linear-quadratic Gaussian regulator problem in one dimension. The real-valued state process X = (Xt; t 2 [0; T ]) is governed by the stochastic di�erential equation dXt = a(t)Xtdt + b(t)utdt + c(t)dBHt ; t 2 [0; T ] ; X0 = x; (0.1) which is as usual interpreted as an integral equation. Here x is a �xed initial condition, BH = (BHt ; t 2 [0; T ]) is a normalized fBm with the Hurst parameter H in [1=2; 1) and the coe�cients a = (a(t); t 2 [0; T ]), b = (b(t), t 2 [0; T ]) and c =

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