Viscosity characterization of the explosion time distribution for diffusions

Abstract

We show that the tail distribution $U$ of the explosion time for a multidimensional diffusion (and more generally, a suitable function $\mathscr{U}$ of the Feynman–Kac type involving the explosion time) is a viscosity solution of an associated parabolic partial differential equation (PDE), provided that the dispersion and drift coefficients of the diffusion are continuous. This generalizes a result of Karatzas and Ruf [Probab. Theory Related Fields (2015) To appear], who characterize $U$ as a classical solution of a Cauchy problem for the PDE in the one-dimensional case, under the stronger condition of local Hölder continuity on the coefficients. We also extend their result to $\mathscr{U}$ in the one-dimensional case by establishing the joint continuity of $\mathscr{U}$. Furthermore, we show that $\mathscr{U}$ is dominated by any nonnegative classical supersolution of this Cauchy problem. Finally, we consider another notion of weak solvability, that of the distributional (sub/super)solution, and show that $\mathscr{U}$ is no greater than any nonnegative distributional supersolution of the relevant PDE.