  We consider stochastic differential systems driven by a Brownian motion and a
Poisson point measure where the intensity measure of jumps depends on the
solution. This behavior is natural for several physical models (such as
Boltzmann equation, piecewise deterministic Markov processes, etc). First, we
give sufficient conditions guaranteeing that the semigroup associated with such
an equation preserves regularity by mapping the space of the of k-times
differentiable bounded functions into itself. Furthermore, we give an explicit
estimate of the operator norm. This is the key-ingredient in a quantitative
Trotter-Kato-type stability result: it allows us to give an explicit estimate
of the distance between two semigroups associated with different sets of
coefficients in terms of the difference between the corresponding infinitesimal
operators. As an application, we present a method allowing to replace " small
jumps " by a Brownian motion or by a drift component. The example of the 2D
Boltzmann equation is also treated in all detail.
