Uniqueness for the martingale problem associated with pure jump processes

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 L defined as Lfx= fx+h-f x-1h ≤11fx ˙hnx,h hd+a xdh, 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 L . Sufficient conditions for existence and uniqueness of the solution to the martingale problem for L are discussed. In the case of a fixed index α, the martingale problem associated with L has a unique solution if :n(x, h)−1: ≤ c(1 ∧ :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 L given by Lfx= fx+h-f x-1h ≤11fx ˙hnx,h hd+a dh. This is the infinitesimal generator of a stable-like process. We show that there exists a unique solution to the martingale problem for L under some mild conditions on n(x, h). We also obtain the transition density estimates for the process generated by the operator M (defined as Mfx= e-iu˙xℓx,u f&d4;u du, where ℓ(x,u) = − j=1n cj:u·ν j:α and νj are unit vectors.