Abstract

In the first chapter of this dissertation, we introduce the martingale problem and present some historical results thereof.

In the second chapter of this dissertation, we consider the operator [special characters omitted] defined as [special characters omitted] where f is a C² function. This is an operator of variable order and the corresponding process is a pure jump process. We consider the martingale problem associated with [special characters omitted]. Sufficient conditions for existence and uniqueness of the solution to the martingale problem for [special characters omitted] are discussed. In the case of a fixed index α, the martingale problem associated with [special characters omitted] has a unique solution if |n(x, h)−1| ≤ c(1[special characters omitted]|h|&epsis;) for a certain positive constant c and a certain positive &epsis;. Transition density estimates for α-stable processes are also obtained as well as estimates on the derivatives of these densities.

In the third chapter of this dissertation, we consider the martingale problem associated with the operator [special characters omitted] given by [special characters omitted] This is the infinitesimal generator of a stable-like process. We show that there exists a unique solution to the martingale problem for[special characters omitted] under some mild conditions on n(x, h). We also obtain the transition density estimates for the process generated by the operator [special characters omitted] (defined as [special characters omitted] where ℓ(x,u) = −[special characters omitted]c_j|u·ν _j|^α and ν_j are unit vectors.