Maximum Principles for Conditional Mean Field Type Control Problems Under Partial and Full Observation with Applications

Motivated by an asset-liability management problem and an optimal investment of N agents under relative performance criteria with common noise, this paper studies optimal control problems of conditional mean field type under partial and full observation. For the partial observation case, we remove the restriction that observation coefficient is uniformly bounded and allow it to grow linearly (unbounded) with respect to state variable, which gives rise to some difficulties in the subsequent analysis. Different with the conventional approximate method (indirect method), we derive a maximum principle with linear observation coefficient adopting a dominated growth rate idea (direct method). As adjoint equation, a conditional mean field backward stochastic differential equation with stochastic Lipschitz coefficients is introduced. Combining backward separation method with state-augmentation technique, a closed form candidate optimal premium strategy of asset-liability management problem is derived. For the full observation case, by virtue of the derived maximum principle and mean field game theory, we investigate a conditional mean-variance portfolio selection problem and obtain the efficient frontier and efficient portfolio explicitly, which is proved to be an \(\epsilon \)-Nash equilibrium of the optimal investment of N agents under relative performance concern with common noise. Some numerical simulations with sound financial interpretations are presented, which verify the effectiveness of our theoretical results.