A Hilbert space method to LQ optimal control of conditional McKean–Vlasov dynamics

In this paper, we employ a Hilbert space approach based on orthogonal decomposition to investigate an optimal control problem associated with a linear conditional McKean–Vlasov dynamic, where the control process is allowed to be adapted to the filtration generated by both common and idiosyncratic noises under random coefficients. By applying the stochastic maximum principle, we derive the optimal feedback control and characterize the corresponding value function in terms of backward stochastic Riccati differential equations. Furthermore, an application to conditional mean–variance portfolio selection in an incomplete market setting is provided.