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 lambda > 0. The utility function is of the form U-p(c) = c(p)/p for p < 1, p not equal 0. We provide a heuristic and a rigorous derivation of the asymptotic expansion of the value function in powers of lambda(1/3). We also obtain a "nearly optimal" strategy, whose utility asymptotically matches the leading terms in the value function.