Portfolio optimization is a well-known problem in mathematical finance concerned with selecting a portfolio which will maximize the expected terminal utility of an investor given today's information and subject to some constraints. It has been studied extensively under various assumptions on the market, including completeness. The assumptions on the market and price processes in this paper are the same as those made in [9]. We find a closed-form formula for a portfolio under which the expected utility is close to optimal when the time horizon is small. As in [9], we work with the problem of portfolio optimization via its associated Hamilton-Jacobi-Bellman (HJB) equation. Specifically, we work with the "marginal HJB equation" given in [9]. We find a classical sub- and super-solution to the marginal HJB equation. A comparison principle argument for a logarithmic transformation of the marginal HJB equation then yields the result.
↧