PDE for the joint law of the pair of a continuous diffusion and its running maximum

TL;DR

This paper derives a PDE characterizing the joint density of a diffusion process's first component and its running maximum, providing a mathematical framework for understanding their joint distribution.

Contribution

It establishes that the joint density of the diffusion's maximum and the process itself solves a specific Fokker-Planck PDE, extending previous results to a multidimensional setting.

Findings

Joint density satisfies a Fokker-Planck PDE

Provides integral expansion for the density

Applicable to multidimensional diffusions

Abstract

Let X be a d-dimensional diffusion and M the running supremum of its first component. In this paper, we show that for any t > 0, the density (with respect to the d + 1-dimensional Lebesgue measure) of the pair (Mt, Xt) is a weak solution of a Fokker-Planck partial differential equation on the closed set {(m, x) $\in$ R d+1, m $\ge$ x 1}, using an integral expansion of this density.