  Given a stochastic dynamical system modelled via stochastic differential
equations (SDEs), we evaluate the safety of the system through
characterizations 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. However, the existing moment
approach suffers from drawbacks which impede its applicability to the analysis
of higher dimensional physical systems. To apply the existing approach, the
dynamics of the systems are limited to polynomials of the state - excluding a
large majority of real world examples. Computational scalability is also poor
as the dimensionality of the state increases, largely due to the combinatorial
growth of the optimization program. In this paper, we propose changes to the
existing moment method to make feasible the safety analysis of higher
dimensional physical systems. The restriction to polynomial dynamics is lifted
by using a state augmentation method which allows one to generate the evolution
equations for a broader class of nonlinear stochastic systems. We then
reformulate the constraints of the optimization to mitigate the computational
limitations associated with an increase in state dimensionality. We employ our
methodology on two example processes to characterize their safety via expected
exit times and demonstrate the ability to handle multi-dimensional physical
systems that were previously unsupported by the existing SDP method of moments.
