The martingale problem for a class of stable-like processes

TL;DR

This paper establishes the existence and uniqueness of solutions to the martingale problem for a class of stable-like processes characterized by a non-local operator with variable coefficients.

Contribution

It proves the well-posedness of the martingale problem for a broad class of stable-like processes with variable coefficients under mild conditions.

Findings

Existence of solutions to the martingale problem for the operator L.

Uniqueness of solutions under mild conditions on A.

Extension of classical stable process theory to variable coefficient cases.

Abstract

Let $\alpha\in (0,2)$ and consider the operator $$L f(x) =\int [f(x+h)-f(x)-1_{(|h|\leq 1)} \nabla f(x)\cdot h] \frac{A(x,h)}{|h|^{d+\alpha}} dh, $$ where the $\nabla f(x)\cdot h$ term is omitted if $\alpha<1$. We consider the martingale problem corresponding to the operator $L$ and under mild conditions on the function $A$ prove that there exists a unique solution.