  We consider a general d-dimensional Levy-type process with killing. Combining
the classical Dyson series approach with a novel polynomial expansion of the
generator A(t) of the Levy-type process, we derive a family of asymptotic
approximations for transition densities and European-style options prices.
Examples of stochastic volatility models with jumps are provided in order to
illustrate the numerical accuracy of our approach. The methods described in
this paper extend the results from Corielli et al. (2010), Pagliarani and
Pascucci (2013) and Lorig et al. (2013a) for Markov diffusions to Markov
processes with jumps.
