On Some Cheap Control Problems for Diffusion Processes

We consider several cases of control problems for diffusion processes when the payoff functional does not depend explicitly on the control. We prove the continuity of the optimal cost function and give a characterization of this cost with a quasi-variational inequality interpreting the problem as limit of an impulse control problem when the cost of impulse tends to zero. Moreover, we show the existence of an optimal control for some particular situations. Introduction. This paper is devoted to the study of the behavior of several kinds of impulse control problems where the fixed cost either tends to zero, or is zero when the impulse is zero. Moreover, some properties of the limit problem are examined, especially existence of an optimal control and characterization of one optimal. We develop the results announced in [19]. The general theory of impulse control leads to a quasi-variational inequality (Q.V.I.) in the stationary case of the form Au + au 0 (Au + au f)(uMu) = 0 for the inventory-like control problem; see Bensoussan and Lions [5,6], Menaldi [17] and Robin [22]. When k -> 0, c = 0, some results have been obtained by Menaldi, Quadrat and Rofman [18] and Menaldi and Rofman [20]. When k = 0, c(£) -» +oo, as £ -> oo and c(|)/£ -* oo as £ -» 0, the one-dimensional case was considered by Vickson [23] for a capacity expansion problem. On the other hand, the limit problem, where c = 0, and k |0, leads formally to a constraint of the form u'x > 0 in the one-dimensional case, or u'x > 0, i = l,...,w, if x E R" and it has been considered for different problems (monotone follower problems) by many authors: Barren and Jensen [1], Bather and Chernoff [2], Borodovskii, Bratus and Chernousko [7], Bratus [8], Chernousko [11], Gorbunov [12] and Karatzas [13, 14], generally in the onedimensional case for special cases (mainly a pure Wiener process), although the problem was already investigated by Chernousko [10] and Benes, Shepp and Received by the editors August 23, 1982. 1980 Mathematics Subject Classification. Primary 93E20, 35J20; Secondary 49A29, 60J60.