On the Infinite Time Horizon Linear-quadratic Regulator Problem under a Fractional Brownian Perturbation

Recently, stochastic models appropriate for long-range dependent phenomena have attracted a great deal of interest and numerous theoretical results and successful applications have been already reported. In particular, several contributions in the literature have been devoted to the extension of the classical theory of continuoustime 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 filtering is now rather well understood (see, e.g., [6–9, 15, 18, 19] and references therein). As far as we know, concerning optimal control problems, it is not yet fully demonstrated. Nevertheless, see [1] for a recent attempt in a general setting and [10] for a complete solution of the simplest linear-quadratic problem on a finite time interval. Here our aim is to illustrate further the actual solvability of control problems by exhibiting an explicit solution for the basic infinite time fractional linear-quadratic regulator problem. 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 ≥ 0) is governed by the stochastic differential equation dXt = aXtdt+ butdt+ dB t , t ≥ 0, X0 = x, (1.1)