PDE for the joint law of the pair of a continuous diffusion and its running maximum
Résumé
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) ∈ R d+1, m ≥ x 1}, using an integral expansion of this density.
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