Finite and mean field games for optimal investment with HARA utility function and the presence of risk-seeking agents

Abstract This study extends the work of Lacker and Zariphopoulou LZ19 by considering the financial market with the presence of both risk-averse and risk-seeking agents. Specifically, the n-agent (finite) and mean field games for optimal investment with the family of the hyperbolic absolute risk aversion (HARA) utility function under relative performance concern/motivation are studied. Several specific forms of the HARA family, including exponential, power, and logarithmic form are investigated. We prove that there exists a unique constant Nash equilibrium and a unique constant mean field equilibrium in both the n-agent and mean field games for the case of strictly concave utility function. For the case of strictly convex utility function, there exists a unique corner solution in these games where agents invest all of their wealth in risky assets (e.g. stock) and invest nothing on riskless assets (e.g. bond). Furthermore, we discuss the qualitative effects of the personal and market coefficients on the optimal investment strategies.