Uniqueness for the martingale problem associated with pure jump processes of variable order

Abstract

Let $L$ be the operator defined on $C^2$ functions by $$Lf(x)=\int [f(x+h)-f(x)-1_{(|h|\le 1)}\nabla f(x)\cdot h]\frac{n(x,h)}{|h{|}^{d+\alpha (x)}}dh.$$ This is an operator of variable order and the corresponding process is of pure jump type. We consider the martingale problem associated with $L$. Sufficient conditions for existence and uniqueness are given. Transition density estimates for $\alpha$-stable processes are also obtained.

References in corpus (1)

• Stochastic differential equations with jumps

Cited by in corpus (5)

• A new technique for proving uniqueness for martingale problems
• Uniform Hölder Estimates on Semigroups Generated by Non-Local Operators of Variable Order
• Uniqueness in law for stable-like processes of variable order
• Regularity results for stable-like operators
• An inequality for a class of Markov processes