Viscosity Characterization of the Explosion Time Distribution for Diffusions

TL;DR

This paper proves that the tail distribution of explosion times for multidimensional diffusions is a viscosity solution of a related PDE, extending previous one-dimensional results and analyzing various solution notions.

Contribution

It generalizes the characterization of explosion time distributions as viscosity solutions from one-dimensional to multidimensional diffusions with continuous coefficients.

Findings

The tail distribution $U$ is a viscosity solution of the associated PDE.

$\mathscr{U}$ is dominated by classical supersolutions of the PDE.

In one dimension, $\mathscr{U}$ is jointly continuous.

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 (2013), 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 H\"older continuity on the coefficients. Furthermore, we show that $\mathscr{U}$ is dominated by any nonnegative classical supersolution of this Cauchy problem, and that $\mathscr{U}$ is the smallest lower-semicontinuous viscosity supersolution of that PDE with an appropriate boundary condition, provided it is a classical solution. We also 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. Finally, we establish the joint continuity of $\mathscr{U}$ in the one-dimensional case.