  This paper investigates optimal consumption in the stochastic Ramsey problem
with the Cobb-Douglas production function. Contrary to prior studies, we allow
for general consumption processes, without any a priori boundedness constraint.
A non-standard stochastic differential equation, with neither Lipschitz
continuity nor linear growth, specifies the dynamics of the controlled state
process. A mixture of probabilistic arguments are used to construct the state
process, and establish its non-explosiveness and strict positivity. This leads
to the optimality of a feedback consumption process, defined in terms of the
value function and the state process. Based on additional viscosity solutions
techniques, we characterize the value function as the unique classical solution
to a nonlinear elliptic equation, among an appropriate class of functions. This
characterization involves a condition on the limiting behavior of the value
function at the origin, which is the key to dealing with unbounded
consumptions. Finally, relaxing the boundedness constraint is shown to
increase, strictly, the expected utility at all wealth levels.
