Portfolio selection and the maximum principle approach in Itô-Liu financial markets

This study applies the maximum principle approach to examine an optimal control problem in Itô-Liu financial markets, also known as uncertain stochastic financial markets. To this end, the Mangasarian sufficient maximum principle approach is proposed and applied within this uncertain stochastic setting. In addition, the Itô representation theorem is extended to the case of uncertain random processes. These concepts are then applied to an optimal portfolio selection problem. The optimal portfolio rule derived via the maximum principle approach agrees with the standard Merton solution obtained via dynamic programming in the purely stochastic setting, thereby validating the maximum principle approach in the context of Itô-Liu financial markets. Numerical examples demonstrate the practical implementation of the developed technique. From these results, we conclude that the maximum principle approach offers a robust means of solving optimal control problems in this setting. The contribution of the study is threefold: first, it proposes a Mangasarian-type sufficient maximum principle approach for continuous-path Itô-Liu processes, avoiding the smoothness requirements of the Hamilton-Jacobi-Bellman (HJB) approach; second, it derives an explicit closed-form optimal portfolio under logarithmic utility and third, it extends the Itô representation theorem to uncertain random processes via a Clark-Ocone-type formula, which is absent in the existing uncertain-stochastic portfolio literature.