Viscosity Characterization of the Value Function of an Investment-Consumption Problem in Presence of an Illiquid Asset

We study a problem of optimal investment/consumption over an infinite horizon in a market consisting of a liquid and an illiquid asset. The liquid asset is observed and can be traded continuously, while the illiquid one can only be traded and observed at discrete random times corresponding to the jumps of a Poisson process. The problem is a nonstandard mixed discrete/continuous optimal control problem, which we face by the dynamic programming approach. The main goal of the paper is the characterization of the value function as unique viscosity solution of an associated Hamilton–Jacobi–Bellman equation. We then use such a result to build a numerical algorithm, allowing one to approximate the value function and so to measure the cost of illiquidity