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

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 (M-t, X-t) is a weak solution of a 1 Fokker-Planck partial differential equation on the closed set {(m, x) e Rd+1, m >= x(1)}, using an integral expansion of this density.