  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.
