  We present a theoretical framework for characterizing incremental stability
of nonlinear stochastic systems perturbed by compound Poisson shot noise and
finite-measure L\'{e}vy noise. For each noise type, we compare trajectories of
the perturbed system with distinct noise sample paths against trajectories of
the nominal, unperturbed system. We show that for a finite number of jumps
arising from the noise process, the mean-squared error between the trajectories
exponentially converge towards a bounded error ball across a finite interval of
time under practical boundedness assumptions. The convergence rate for shot
noise systems is the same as the exponentially-stable nominal system, but with
a tradeoff between the parameters of the shot noise process and the size of the
error ball. The convergence rate and the error ball for the L\'{e}vy noise
system are shown to be nearly direct sums of the respective quantities for the
shot and white noise systems separately, a result which is analogous to the
L\'{e}vy-Khintchine theorem. We demonstrate our results using several numerical
case studies.
