  Given a stochastic dynamical system modelled via stochastic differential
equations (SDEs), we evaluate the safety of the system through
characterisations of its exit time moments. We lift the (possibly nonlinear)
dynamics into the space of the occupation and exit measures to obtain a set of
linear evolution equations which depend on the infinitesimal generator of the
SDE. Coupled with appropriate semidefinite positive matrix constraints, this
yields a moment-based approach for the computation of exit time moments of SDEs
with polynomial drift and diffusion dynamics. To extend the capability of the
moment approach, we propose a state augmentation method which allows us to
generate the evolution equations for a broader class of nonlinear stochastic
systems and apply the moment method to previously unsupported dynamics. In
particular, we show a general augmentation strategy for sinusoidal dynamics
which can be found in most physical systems. We employ the methodology on an
Ornstein-Uhlenbeck process and stochastic spring-mass-damper model to
characterise their safety via their expected exit times and show the additional
exit distribution insights that are afforded through higher order moments.
