[Paper Review] Distribution of the time to explosion for one-dimensional diffusions

Ioannis Karatzas, Johannes Ruf|arXiv (Cornell University)|Mar 24, 2013

Stochastic processes and financial applications40 references4 citations

TL;DR

This paper provides a comprehensive analysis of the distribution of explosion times for one-dimensional diffusions, establishing that the tail probability $\mathbb{P}_\xi(S > t)$ is the minimal nonnegative solution to a parabolic Cauchy problem. It connects the explosion time distribution to expectations of nonnegative local martingales and offers analytic characterizations of Feller’s explosion criterion through PDEs, with explicit solutions in key examples like Bessel processes and Bessel bridges.

ABSTRACT

We study the distribution of the time to explosion for one-dimensional diffusions. We relate this question to computing the expectations of suitable nonnegative local martingales, and to the distributions of related diffusions with unit variance. Moreover, we characterize the distribution function of the time to explosion as the minimal solution to a certain Cauchy problem for an appropriate parabolic differential equation; this leads to alternative characterizations of Feller's criterion for explosions. We discuss in detail several examples for which it is possible to obtain analytic expressions for the corresponding distribution of the time to explosion, using the methodologies developed in the paper.

Motivation & Objective

• To characterize the distribution of the explosion time for one-dimensional diffusions, a topic largely unexplored despite Feller’s foundational work on explosion criteria.
• To establish a connection between the explosion time distribution and expectations of nonnegative local martingales, generalizing McKean’s result.
• To show that the tail probability $\mathbb{P}_\xi(S > t)$ is the minimal solution to a parabolic Cauchy problem, offering a new analytic framework for studying explosion.
• To investigate analytic properties such as continuity, strict positivity, and full support of the explosion time distribution.
• To provide explicit analytical expressions for the explosion time distribution in key examples, including Bessel processes and Bessel bridges, using the developed methodology.

Proposed method

• Uses the theory of one-dimensional diffusions and pathwise solutions of SDEs to model the dynamics of the process before explosion.
• Applies Girsanov’s theorem and the concept of nonnegative local martingales to relate the explosion time distribution to expectations under a change of measure.
• Characterizes the tail probability $\mathbb{P}_\xi(S > t)$ as the minimal nonnegative solution to a parabolic partial differential equation (PDE) of the form $\mathcal{L}u = 0$, where $\mathcal{L}$ is the generator of the diffusion.
• Derives an alternative characterization of the Laplace transform of the explosion time distribution via an ordinary differential equation (ODE) in the scale function framework.
• Employs the Lamperti transformation and the Feller test for explosions to analyze boundary behavior and explosion conditions.
• Establishes uniqueness in distribution for stopped diffusions via time-changed Brownian motion and inverse time change arguments, ensuring robustness of the solution framework.

Experimental results

Research questions

• RQ1 How can the distribution of the explosion time for a one-dimensional diffusion be characterized analytically beyond Feller’s criterion?
• RQ2 What is the precise relationship between the explosion time distribution and expectations of nonnegative local martingales?
• RQ3 Can the tail probability $\mathbb{P}_\xi(S > t)$ be characterized as a solution to a PDE, and if so, what kind of solution is it?
• RQ4 Under what conditions does the explosion time distribution have full support or exhibit strict positivity?
• RQ5 What explicit expressions can be derived for the explosion time distribution in specific diffusions such as Bessel processes?

Key findings

• The tail probability $\mathbb{P}_\xi(S > t)$ is the minimal nonnegative solution to a parabolic Cauchy problem associated with the generator of the diffusion.
• The explosion time distribution is fully characterized via the minimal solution of a PDE, offering a new analytic tool to study explosion behavior.
• The Laplace transform of the explosion time distribution satisfies an ordinary differential equation in the scale function, enabling alternative analytical treatment.
• Feller’s criterion for explosion is reinterpreted as a condition on the minimal solution of the PDE, providing a PDE-based alternative to the classical scale function condition.
• For Bessel processes of dimension $\delta \in (0,2)$, the paper derives explicit expressions for the distribution of the explosion time using the developed PDE and local martingale methods.
• The distribution of the explosion time for the Bessel bridge is shown to be expressible in terms of the first-passage time of a Brownian motion with drift, leveraging the time-changed Brownian motion representation.