Portfolio Optimization Problems

We propose a optimal portfolio problem where the underlying is driven by the factor model with delay feature in order to describe the interaction with time delay among different financial markets. The delay phenomenon can be recognized as the integral type and the pointwise type. Due to the delay leading to the non-Markovian structure, we obtain the optimal strategy through the coupled forward and backward stochastic differential equations (FBSDEs). The existence and uniqueness of the coupled FBSDEs are also studied. We analyze three particular cases where the corresponding FBSDEs can be solved explicitly. In addition, we consider the problem of optimal investment by an insurer where the claim size is driven by Brownian motion. The optimal strategy is obtained using the coupled FBSDEs. Finally, we analyze the model of high frequency trading in order to obtain the optimal ask quote based on the feature observed in financial markets under relative performance. The Nash equilibria are solved using the coupled Hamilton Jacobi Bellman (HJB) equations.