  We present a theoretical framework for characterizing incremental stability
of nonlinear stochastic systems perturbed by additive noise of two types:
compound Poisson shot noise and bounded-measure L\'evy noise. For each type, we
show that trajectories of the system with distinct initial conditions and noise
sample paths exponentially converge towards a bounded error ball of each other
in the expected mean-squared sense under certain boundedness assumptions. The
practical utilities of this study include the model-based design of stochastic
controllers/observers that are able to handle a much broader class of noise
than Gaussian white. We demonstrate our results using three case studies.
