Asymptotic analysis for optimal investment in finite time with transaction costs

We consider an agent who invests in a stock and a money market account with the goal of maximizing the utility of her investment at the final time T in the presence of a proportional transaction cost λ > 0. The utility function is of the form U_p(c) = c^p/p for p < 1, p ≠ 0. We provide a heuristic and a rigorous derivation of the asymptotic expansion of the value function in powers of λ^1/3. We also obtain a "nearly optimal" strategy, whose utility asymptotically matches the leading terms in the value function.