“…Among others, these include dynamic programming techniques (see, e.g., [23], [29], [35] and [38]), stochastic first-order conditions and the Bank-El Karoui's Representation Theorem [4] (see, e.g., [5], [15], [24] and [44]), the transformation method of [7] in the case of one-dimensional problems, and the analytical study of non linear PDEs with gradient constraints (see for example [46] and [47]). The introduction of a stochastic investment cost Y t is very natural from the point of view of economic modelling (see e.g.…”