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February 2017 Viscosity characterization of the explosion time distribution for diffusions
Yinghui Wang
Bernoulli 23(1): 503-521 (February 2017). DOI: 10.3150/15-BEJ752

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.

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Yinghui Wang. "Viscosity characterization of the explosion time distribution for diffusions." Bernoulli 23 (1) 503 - 521, February 2017. https://doi.org/10.3150/15-BEJ752

Information

Received: 1 June 2015; Revised: 1 July 2015; Published: February 2017
First available in Project Euclid: 27 September 2016

zbMATH: 1361.60069
MathSciNet: MR3556781
Digital Object Identifier: 10.3150/15-BEJ752

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

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Vol.23 • No. 1 • February 2017
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