  We extend the work on optimal investment and consumption of a population
considered in [2] to a general stochastic setting over a finite time horizon.
We incorporate the Cobb-Douglas production function in the capital dynamics
while the consumption utility function and the drift rate in the population
dynamics can be general, in contrast with [2, 30, 31]. The dynamic programming
formulation yields an unconventional nonlinear Hamilton-Jacobi-Bellman (HJB)
equation, in which the Cobb-Douglas production function as the coefficient of
the gradient of the value function induces the mismatching of power rates
between capital and population. Moreover, the equation has a very singular
term, essentially a very negative power of the partial derivative of the value
function with respect to the capital, coming from the optimization of control,
and their resolution turns out to be a complex problem not amenable to
classical analysis. To show that this singular term, which has not been studied
in any physical systems yet, does not actually blow up, we establish new
pointwise generalized power laws for the partial derivative of the value
function. Our contribution lies in providing a theoretical treatment that
combines both the probabilistic approach and theory of partial differential
equations to derive the pointwise upper and lower bounds as well as energy
estimates in weighted Sobolev spaces. By then, we accomplish showing the
well-posedness of classical solutions to a non-canonical parabolic equation
arising from a long-lasting problem in macroeconomics.
