  Management of the portfolios containing low liquidity assets is a tedious
problem. The buyer proposes the price that can differ greatly from the paper
value estimated by the seller, the seller, on the other hand, can not liquidate
his portfolio instantly and waits for a more favorable offer. To minimize
losses in this case we need to develop new methods. One of the steps moving the
theory towards practical needs is to take into account the time lag of the
liquidation of an illiquid asset. This task became especially significant for
the practitioners in the time of the global financial crises. Working in the
Merton's optimal consumption framework with continuous time we consider an
optimization problem for a portfolio with an illiquid, a risky and a risk-free
asset. While a standard Black-Scholes market describes the liquid part of the
investment the illiquid asset is sold at a random moment with prescribed
liquidation time distribution. In the moment of liquidation it generates
additional liquid wealth dependent on illiquid assets paper value. The investor
has the logarithmic utility function as a limit case of a HARA-type utility.
Different distributions of the liquidation time of the illiquid asset are under
consideration - a classical exponential distribution and Weibull distribution
that is more practically relevant. Under certain conditions we show the
existence of the viscosity solution in both cases. Applying numerical methods
we compare classical Merton's strategies and the optimal consumption-allocation
strategies for portfolios with different liquidation-time distributions of an
illiquid asset.
