IMPACT OF TIME ILLIQUIDITY IN A MIXED MARKET WITHOUT FULL OBSERVATION

We study a problem of optimal investment/consumption over an innite 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 , is observed at the trading dates, and is partially observed between two dierent 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 sucient regularity of the value function, we give a verication 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 verication 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 aected by the degree