#### Title

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 h^{d+a x}dh, where *f* is a *C*^{2} 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 h^{d+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˙x}ℓx,u f&d4;u du, where ℓ(*x,u*) = − j=1

- n

*c*:

_{j}*u*·ν

*:*

_{ j}^{α}and ν

*are unit vectors.^*

_{j}#### Recommended Citation

Tang, Huili, "Uniqueness for the martingale problem associated with pure jump processes" (2006). *Doctoral Dissertations*. AAI3236152.

http://opencommons.uconn.edu/dissertations/AAI3236152