  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.
