  We study a problem of optimal investment/consumption over an infinite horizon
in a market consisting of two possibly correlated assets: one liquid and one
illiquid. The liquid asset is observed and can be traded continuously, while
the illiquid one can be traded only at discrete random times corresponding to
the jumps of a Poisson process with intensity $\lambda$, is observed at the
trading dates, and is partially observed between two different trading dates.
The problem is a nonstandard mixed discrete/continuous optimal control problem
which we face by the dynamic programming approach. When the utility has a
general form we prove that the value function is the unique viscosity solution
of the HJB equation and, assuming sufficient regularity of the value function,
we give a verification theorem that describes the optimal investment strategies
for the illiquid asset. In the case of power utility, we prove the regularity
of the value function needed to apply the verification theorem, providing the
complete theoretical solution of the problem. This allows us to perform
numerical simulation, so to analyze the impact of time illiquidity in this
mixed market and how this impact is affected by the degree of observation.
