Uniqueness for the martingale problem associated with pure jump processes

Date of Completion

January 2006

Keywords

Mathematics

Degree

Ph.D.

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 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=1

    n
cj:u·ν j:α and νj are unit vectors.^

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