Impact of time illiquidity in a mixed market without full observation

TL;DR

This paper analyzes optimal investment and consumption strategies in a market with one liquid and one illiquid asset, where the illiquid asset is only partially observed and traded at random times, using advanced control and PDE methods.

Contribution

It develops a theoretical framework for a mixed discrete/continuous control problem with partial observations, providing existence, uniqueness, and regularity results for the value function.

Findings

Impact of time illiquidity quantified through numerical simulations

Optimal strategies characterized for general utility functions

Regularity results enable complete solution for power utility case

Abstract

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.