Separation Principle in the Fractional Gaussian Linear-quadratic Regulator Problem with Partial Observation

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. Mathematics Subject Classification. 93E11, 93E20. 60G15, 60G44. Received July 21, 2006. Revised January 8, 2007. 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 basic problems in prediction, parameter estimation, filtering and control is now rather well understood (see, e.g., [1,6–10,15,16], and references therein). Nevertheless, as far as we know, it is not yet demonstrated that optimal control problems can also be handled for fractional stochastic systems which are only partially observable. So, our aim here is to illustrate the actual solvability of such 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 processes X = (Xt, t ≥ 0) and Y = (Yt, t ≥ 0), representing the state dynamic and the available observation record, respectively, are governed by the following linear system of stochastic