On the Lack of Optimal Classical Stochastic Controls in a Capacity Expansion Problem

The stochastic control problem of a firm aiming to optimally expand the production capacity, through irreversible investment, in order to maximize the expected total profits on a finite time interval has been widely studied in the literature when the firm’s capacity is modeled as a controlled Itˆo process in which the control enters additively and it is a general nondecreasing stochastic process, possibly singular as a function of time, representing the cumulative investment up to time t. This note proves that there is no solution when the problem falls in the so-called classical control setting; that is, when the control enters the capacity process as the rate of real investment, and hence the cumulative investment up to time t is an absolutely continuous process (as a function of time). So, in a sense, this note explains the need for the larger class of nondecreasing control processes appearing in the literature