Continous time optimal stochastic growth: local martingales, transversality and existence

This lengthy paper extends the author's work on optimal planning of consumption versus capital accumulation to stochastic versions of traditional continuous-time one­sector growth models. Risk is assumed to be exogenous but is otherwise specified in a very general form. An optimal plan is characterised by means of local martingale conditions for shadow prices and transversality conditions at infinity. The definitions of these conditions involve sequences of random stopping times, and various choices of these times which are of economic interest are considered. For example, assumptions are given which allow the stopping times to be chosen as clock times, so that the local martingale is a true martingale and the expected capital value tends to zero as clock time tends to infinity. The possibility of making random time changes so as to replace ­local by true martingale conditions for an optimum is also considered. Separately, conditions for the existence of an optimum are obtained.