Weak uniqueness for the PDE governing the joint law of a diffusion and its running supremum

In a previous work [8], it was shown that the joint law of a diffusion process and the running supremum of its first component is absolutely continuous, and that its density satisfies a non standard weak partial differential equation (PDE). In this paper, we establish the uniqueness of the solution to this PDE, providing a more complete understanding of the system’s behavior and further validating the approach introduced in [8].