Optimal investment strategies under the relative performance in jump-diffusion markets

We work on a portfolio management problem for one agent and a large group of agents under relative performance concerns in jump-diffusion markets with the CRRA utility function. Herein, we define two wealth dynamics: the agent's and the group's wealth. We measure the performances of both the agent and the group with preferences linked to the group performance. Therefore, we have stochastic optimal control problems for both the representative agent and the group to determine what the group does and the agent's optimal proportion in the portfolio relative to the group's performance. Further, our framework assumes that the agent's performance does not affect the group, while the group affects the agent's utility. Moreover, we investigate special cases where all agents in the market are homogeneous in their risk aversion and relative performances. We explore the qualitative behavior of the agent and show some numerical results depending on her relative performance consideration and risk tolerance degree.