Optimal Investment In Incomplete Markets

In this paper, we mainly study the problem of the optimal investment in incomplete markets. Since Merton first studied this problem of continuous-time models in last 70s, many learners devoted in this study. To this day, the theorems of existence and uniqueness for optimal investment virtually settled, which didn't allow negative wealth. But the Legendre-transform, an important parts of duality methods, learners only gave the formula, not concrete courses. We first consider the classical case where the underlying probability space is finite. In this setting, applying the minimax thoerem, we get the crucial role of Legendre-transform, then we prove an existence and uniqueness theorem for the optimal investment and its relation to the dual problem, and pass this result to the general case of infinite probability space. Finally, we study the optimal problem of an agent who, in addition to an initial capital, receives random endowments , when negative wealth allowed. And then we obtain an existence and uniqueness theorem correspondingly.