Optimal stationary linear control of the Wiener process

Abstract

In the present paper, we consider the following stochastic control problem: to minimize the average expected total cost $$J(x,u) = \mathop {\lim \inf }\limits_{T \to \infty } (1/T)E_x^u \int_0^T {\left[ {\phi (\xi _t ) + |u_t (\xi )|} \right]} dt,$$ 〈subject to $$d\xi _t = u_1 (\xi )dt + dw_t , \xi _0 = x, |u| \leqslant 1,$$ (wt) a Wiener process, with all measurable functions on the past of the state process {ξs;s≤t} and bounded by unity, admissible as controls. It is proved that, under very mild conditions on the running cost function φ(·), the optimal law is of the form $$\begin{gathered} u_t^* (\xi ) = - sign\xi _t , |\xi _t | > b, \hfill u_t^* (\xi ) = 0, |\xi _t | > b. \hfill \end{gathered} $$ The cutoff pointb and the performance rate of the optimal lawu* are simultaneously determined in terms of the function φ(·) through a simple system of integrotranscendental equations.