Optimal discounted linear control of the wiener process

The following stochastic control problem is considered: to minimize the discounted expected total cost $$J(x;u) = E\int_0^\infty {\exp ( - at)\phi } (x_l ) + |u_l (x)|]dt,$$ subject todx _t =u _t (x)dt+dw _t ,x ₀=x, |u _t |≤1, (w _t ) a Wiener process, α>0. All bounded by unity, measurable, and nonanticipative functionalsu _t (x) of the state processx _t are admissible as controls. It is proved that the optimal law is of the form $$\begin{gathered} u_t^* (x) = - 1,x_t > b, \hfill \\ u_t^* (x) = 0,|x_t | \leqslant b, \hfill \\ u_t^* (x) = 1,x_t< - b, \hfill \\ \end{gathered}$$ for some switching pointb > 0, characterized in terms of the function ø(·) through a transcendental equation.