Singular control problems in bounded intervals

Abstract

We consider optimal control problems for one-dimensional diffusion processes y _x (t) = y ^v _x (t), solutions dy _x (t) = g(y _x (t) dt + σ(y _x (t)(dw) + dv _t with y _x(0) = x& isinv;[a,b], the control processes v _t are increasing, positive, and adapted. Several types of expected cost structures associated with each policy v(.) are adopted, e.g. discounted cost, long term average cost and time average cost. Our work is related to [2,6,12,14,16 and 21], where diffusions are allowed to evolve in the whole space, and to [13] and [20], where diffusions evolve only in bounded regions. We shall present some analytic results about value functions, mainly their characterizations, by simple dynamic programming arguments. Several simple examples are explicitly solved to illustrate the singular behaviour of our problems