On the reconstruction of diffusions from first-exit time distributions

TL;DR

This paper demonstrates how to reconstruct the drift or diffusion coefficients of a stochastic diffusion process from first-exit time distributions by transforming the problem into a Sturm-Liouville inverse problem, with unique reconstruction under certain conditions.

Contribution

It introduces a method to reconstruct either the drift or diffusion coefficient of a diffusion process from first-exit time data using Sturm-Liouville theory, highlighting conditions for uniqueness.

Findings

Either drift or diffusion can be uniquely reconstructed if both are known in half the domain.

Reconstruction is possible from first-exit time distributions via Sturm-Liouville inverse problems.

Additional measurements are necessary for full reconstruction when both coefficients are unknown.

Abstract

This paper explores the reconstruction of drift or diffusion coefficients of a scalar stochastic diffusion processes as it starts from an initial value and reaches, for the first time, a threshold value. We show that the distribution function derived from repeated measurements of the first-exit times can be used to formally partially reconstruct the dynamics of the process. Upon mapping the relevant stochastic differential equations (SDE) to the associated Sturm-Liouville problem, results from Gelfand and Levitan \cite{GelLev-51} can be used to reconstruct the potential of the Schr\"{o}dinger equation, which is related to the drift and diffusion functionals of the SDE. We show that either the drift or the diffusion term of the stochastic equation can be uniquely reconstructed, but only if both the drift and diffusion are known in at least half of the domain. No other information can be uniquely reconstructed unless additional measurements are provided. Applications and implementations of our results are discussed.