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

Laure Coutin (1) , Monique Pontier (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

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.

Mots clés: Diffusion process; partial differential equation; running supremum process; joint law

Domaines: Probabilités [math.PR]

Laure Coutin, Monique Pontier. PDE for the joint law of the pair of a continuous diffusion and its running maximum. Journal of Applied Probability, In press, 60 (1), 43 p. ⟨hal-03925172⟩