Joint distribution of a Lévy process and its running supremum.

Let X be a jump-diffusion process and X* its running supremum. In this paper we first show that for any t > 0, the law of the pair (X-t*, X-t) has a density with respect to the Lebesgue measure. This allows us to show that for any t > 0, the law of the pair formed by the random variable X-t and the running supremum X-t* of X at time t can be characterized as a weak solution of a partial differential equation concerning the distribution of the pair (X-t*, X-t). Then we obtain an expression of the marginal density of X-t* for all t > 0.