Some singular control problem with long term average criterion

Abstract

in this paper we consider an optimal control problem for a diffusion process y(t)=y x v solution of dy(t)=g(y)dt + σ(y)dwt+dvt, where vt is an increasing positive adapted process, with the long term average cost $$J\left( v \right) = \mathop {\lim }\limits_{T \uparrow \infty } \inf \frac{1}{T}E\smallint _0^T f\left( {y_x^v \left( t \right)} \right)dt.$$ The paper gives, for one dimensional processes, existence result for an optimal control in relation with a reflected diffusion process as in Karatzas [16], and characterization of the optimal cost. Moreover, the asymptotic analysis of the discounted cost problem is carried out.