We consider a basic model of two-stage optimal decision making involving pure
information learning beforehand and dynamic consumption afterwards: in stage-1
from initial time to a chosen stopping time, the individual investor has access
to full market information and simply updates the underlying stock and
mean-reverting drift processes by paying information costs; in stage-2 starting
from the chosen stopping time, the investor terminates the costly information
acquisition while the public stock prices are still available and free.
Therefore, during stage-2, the investor starts the investment and consumption
based on previous full information and the dynamic partial observations after
the stopping time. Moreover, the investor adopts the habit formation
preference, in which the past consumption affects his current decisions.
Mathematically speaking, we formulate a composite optimal starting and control
problem, in which the exterior problem is to determine the best time to
initiate the investment-consumption decisions and the interior problem becomes
a finite time stochastic control problem with partial information. The value
function of the composite problem is characterized as the unique viscosity
solution of some variational inequalities.