We will report work in progress on the problem of constructing a solution to the martingale problem with respect to a Lévy-type operator in which the Lévy measure term is x-dependent, with x being the argument of the function on which the operator acts. The x-dependence in the Lévy measure term is not assumed to be in the form of an image measure of a standard Lévy measure by an x-dependent jump-size function.
Results on the existence and uniqueness for this form of martingale problem are available in the literature; however, we are interested in the possible construction of solutions. We will present an approach to construct such a solution in the case where, for each x, the corresponding Lévy measure is of finite mass. However, it is known in literature that there exists a simple approach to construct solutions under certain conditions which allow, in particular, for cases where the Lévy measures are not necessarily of finite mass. We are in the process of trying make use of our approach to treat the problem in the case of more general Lévy measures than those with finite mass. Some issues in this direction will be discussed is this talk.
This is joint work with Mark H. A. Davis.