Existence and regularity of law density of a pair (diffusion, first component running maximum)

摘要

Let X be a continuous d-dimensional diffusion process and M the running supremum of the first component. We show that, for all t > 0, the law of the (d+1) random vector (M-t, X-t) admits a density with respect to the Lebesgue measure using Malliavin's calculus. In case d = 1 we prove the regularity of this density. (C) 2019 Elsevier B.V. All rights reserved.