Remarks on optimal strategies to utility maximizations in continuous time incomplete markets

Abstract

In continuous time diffusion models, the optimal strategies to utility maximizations can be obtained by solving a certain partial differential equation. In this paper, we give another proof of this fact in an incomplete market without using the well-known fictitious security arguments. Since we avoid using the fictitious security arguments, we can apply our method to the situations when the markets cannot be completed. We provide an example of such cases where the asset price follows a simple jump process with unpredictable jump sizes and see that we can derive the equation which determines the optimal strategy as usual.