Optimal portfolio of low liquid assets with a log-utility function

In the real market an asset is not completely liquid. An investor should plan a strategy on the grounds that an asset cannot always be traded. In this paper we consider the classical Merton wealth problem, but the risky asset is not completely liquid. The liquidity is represented by the success rate of the trade and the investor can trade the asset at distributed exponentially random times. We find the value function and exhibit a procedure for an asymptotic expansion of the optimal strategy. Further we reveal some characteristics of the optimal strategy by a numerical analysis.