Existence of densities and atoms for the running maximum of time-inhomogeneous jump diffusions

Abstract

We prove absolute continuity of the running maximum $X_T^{\ast }={\sup }_{0\le s\le T}{X}_s$ of one-dimensional time-inhomogeneous Lévy--Itô diffusions driven by a Brownian motion and an independent non-truncated pure-jump Lévy process. Using Bismut's directional Malliavin calculus on the Wiener--Poisson space together with the running-maximum criteria of Song--Xie and Nakagawa--Suzuki, we reduce the problem to constructing an admissible direction $\Theta$. The key is to ensure that the directional derivative $D_{\Theta }{X}_t$ is strictly positive for all $t\in (0,T]$. We give explicit directions in two regimes. In the uniformly elliptic case with time-inhomogeneous coefficients, a purely Brownian perturbation yields an explicit positive integral representation for $D_{\Theta }{X}_t$, and hence $X_T^{\ast }$ admits a density without truncating the jump component. In a Brownian-degenerate pure-jump model with a bounded deterministic time-dependent jump weight $\kappa (t)$ that may vanish on subintervals, we prove absolute continuity under the minimal nondegeneracy-in-time condition ${\int }_0^t\kappa (s{)}^{2}ds>0$ for every $t>0$ and infinite activity of the Lévy measure. A weighted Poisson positivity lemma is the key new input. Finally, we show that silent initial intervals can create atoms and we derive an explicit atom--density decomposition.