Optimal consumption policies in illiquid markets

TL;DR

This paper analyzes optimal consumption strategies in illiquid markets, deriving smoothness of value functions, proving existence of optimal controls, and providing numerical illustrations of strategies between trading dates.

Contribution

It establishes smoothness of value functions and characterizes optimal controls in illiquid markets, extending previous models with new theoretical and numerical insights.

Findings

Existence of optimal consumption and portfolio strategies.

Smoothness of value functions in the liquidity risk model.

Numerical illustrations of consumption behavior between trading dates.

Abstract

We investigate optimal consumption policies in the liquidity risk model introduced in Pham and Tankov (2007). Our main result is to derive smoothness results for the value functions of the portfolio/consumption choice problem. As an important consequence, we can prove the existence of the optimal control (portfolio/consumption strategy) which we characterize both in feedback form in terms of the derivatives of the value functions and as the solution of a second-order ODE. Finally, numerical illustrations of the behavior of optimal consumption strategies between two trading dates are given.