  The mean first exit time and escape probability are utilized to quantify
dynamical behaviors of stochastic differential equations with non-Gaussian
alpha-stable type Levy motions. Both deterministic quantities are characterized
by differential-integral equations(i.e.,differential equations with non local
terms) but with different exterior conditions. The non-Gaussianity of noises
manifests as nonlocality at the level of mean exit time and escape probability.
An objective of this paper is to make mean exit time and escape probability as
efficient computational tools, to the applied probability community, for
quantifying stochastic dynamics. An accurate numerical scheme is developed and
validated for computing the mean exit time and escape probability. Asymptotic
solution for the mean exit time is given when the pure jump measure in the Levy
motion is small.
  From both the analytical and numerical results, it is observed that the mean
exit time depends strongly on the domain size and the value of alpha in the
alpha-stable Levy jump measure. The mean exit time can measure which of the two
competing factors in alpha-stable Levy motion, i.e. the jump frequency or the
jump size, is dominant in helping a process exit a bounded domain. The escape
probability is shown to vary with the underlying vector field(i.e.,drift). The
mean exit time and escape probability could become discontinuous at the
boundary of the domain, when the process is subject to certain deterministic
potential and the value of alpha is in (0,1).
