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

TL;DR

This paper proves the uniqueness of solutions to a non-standard PDE that describes the joint law of a diffusion process and its running supremum, enhancing understanding of their probabilistic behavior.

Contribution

It establishes the weak uniqueness of the PDE governing the joint law of a diffusion and its supremum, building on previous work that identified the PDE and its properties.

Findings

Uniqueness of the PDE solution is proven.

The result confirms the well-posedness of the joint law characterization.

Supports the validity of the PDE approach for such stochastic processes.

Abstract

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].