Linear Quadratic Optimal Control Problems for Conditional Mean‐Field Stochastic Differential Equations Under Partial Information

ABSTRACT This paper centers on a kind of linear quadratic stochastic optimal control problem driven by conditional mean‐field stochastic differential equations under partial information. In this context, the cost functional is permitted to be indefinite. At the outset, we present a broad overview of optimal control with the aid of the adjoint equation. However, the Hamiltonian system poses a challenge as it incorporates two distinct conditional expectations, making decoupling unattainable. To tackle this, we extract and analyze three representative cases derived from practical problems, discussing each case separately. We find that, in any case, the uniform convexity of the cost functional ensures the existence of a unique optimal control with a state feedback form for the problem, which is a weaker assumption compared to the standard one. Finally, we apply the obtained results to address specific issues raised in the initial motivations of this paper. These applications demonstrate the practical relevance and effectiveness of our theoretical findings in addressing real‐world challenges in the field of stochastic optimal control.