LQG Differential Stackelberg Game Under Nested Observation Information Patterns

We investigate the linear quadratic Gaussian–Stackelberg game under a class of nested observation information patterns. The follower uses its observation data to design its strategy, whereas the leader implements its strategy using global observation data. We show that the solution requires solving a new type of forward–backward stochastic differential equation, whose drift components contain two conditional expectation terms associated with the adjoint variables. We then propose a method to find the functional relations between each adjoint pair, i.e., each pair formed by an adjoint variable and the conditional expectation of its associated state. The proposed method follows a layered pattern. More precisely, in the inner layer, we seek the functional relation for the adjoint pair under the sigma-subalgebra generated by follower's observation information, and in the outer layer, we look for the functional relation for the adjoint pair under the sigma-subalgebra generated by leader's observation information. Our result shows that the optimal open-loop solution admits an explicit feedback type representation.