  We present a scalable underapproximation of the terminal hitting time
stochastic reach-avoid probability at a given initial condition, for
verification of high-dimensional stochastic LTI systems. While several
approximation techniques have been proposed to alleviate the curse of
dimensionality associated with dynamic programming, these techniques are
limited and cannot handle larger, more realistic systems. We present a scalable
method that uses Fourier transforms to compute an underapproximation of the
reach-avoid probability for systems with disturbances with arbitrary
probability densities. We characterize sufficient conditions for
Borel-measurability of the value functions. We exploit fixed control sequences
parameterized by the initial condition (an open-loop control policy) to
generate the underapproximation. For Gaussian disturbances, the
underapproximation can be obtained using existing efficient algorithms by
solving a convex optimization problem. Our approach produces non-trivial lower
bounds and is demonstrated on a chain of integrators with 40 states.
