  We introduce a doubly stochastic method for performing material failure
theory based forecasts of volcanic eruptions. The method enhances the well
known Failure Forecast Method equation, introducing a new formulation similar
to the Hull-White model in financial mathematics. In particular, we incorporate
a stochastic noise term in the original equation, and systematically
characterize the uncertainty. The model is a stochastic differential equation
with mean reverting paths, where the traditional ordinary differential equation
defines the mean solution. Our implementation allows the model to make
excursions from the classical solutions, by including uncertainty in the
estimation. The doubly stochastic formulation is particularly powerful, in that
it provides a complete posterior probability distribution, allowing users to
determine a worst case scenario with a specified level of confidence. We apply
the new method on historical datasets of precursory signals, across a wide
range of possible values of convexity in the solutions and amounts of
scattering in the observations. The results show the increased forecasting
skill of the doubly stochastic formulation of the equations if compared to
statistical regression.
