Hamilton-Jacobi-Bellman Equations in the Wasserstein Space for the Optimal Control of the Kushner-Stratonovich Equation

This paper develops a comparison theorem for viscosity solutions of a new class of Hamilton-Jacobi-Bellman (HJB) equations, which is used to solve the separated problem governed by the K-S equation in the Wasserstein space. A distinctive feature of these HJB equations is the simultaneous presence of variational and Lions derivatives, an inevitable consequence of a nonzero observation function. Moreover, the presence of state-dependent correlated noise adds further complexity to the analysis, making the proof of the comparison theorem more challenging. The core proof strategy for the comparison theorem is to introduce a novel adaptation of the doubling variables argument, specifically tailored to tackle the challenges posed by the Wasserstein space. To this end, we construct an entirely new bivariate functional that combines viscosity sub/supsolutions, a smooth gauge-type function, and two Gaussian regularized entropy penalizations. The gauge-type function compensates for the non-differentiability (in the Lions sense) of the 2-Wasserstein distance, while the entropy penalizations ensure that the maximal point of the functional is well-behaved. Another major contribution of this paper is the derivation of the second-order variational and Lions derivatives of the gauge-type function and entropy functional, which are crucial for dealing with the second-order HJB equation. Meanwhile, the simple structure and pleasing regularity of these derivatives make the methodologies developed in this paper applicable within a wider range of theoretical settings. This paper gives the first result concerning the uniqueness of viscosity solutions for second-order HJB equations with variational derivatives in the Wasserstein space.