Optimal discounted control for a continuous time inventory model

This paper is concerned with the following linear stochastic control problem: Minimize the discounted total cost $$J(x; u) = E{_x} \left {\int_0^\infty {\exp - \alpha t]\{ \phi (x{_t} ) + |u{_t} |\} } dt} \right]$$ over all measurable and nonanticipative control processes (u _t ), subject todx _t =u _t dt+dw _t ,x(0)=x, |u _t |≤1. This problem is analyzed using a discretization technique. The results obtained extend those derived in Ref. 1 and some of those derived in Ref. 2.