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

When an asset is completely liquid, an investor can realize his desirable strategy. But when the asset is not sufficiently liquid, the investor cannot trade the asset continuously and his strategy is restricted. He has to consider the risk of the failure of the trade. In this paper a risky asset is traded at the random times and an investor has a power utility function. In this situation we solve an optimal portfolio problem. We propose an asymptotic expansion of the optimal strategy. Further we discuss convergence of the value function when the asset becomes liquid.