The martingale problem for geometric stable-like processes

TL;DR

This paper establishes the well-posedness of the martingale problem for a class of pure-jump Lévy-type operators with kernels resembling geometric stable processes, extending the mathematical understanding of such jump processes.

Contribution

It proves the well-posedness of the martingale problem for a broad class of Lévy-type operators with kernels similar to geometric stable processes, including new cases with slowly varying functions.

Findings

Martingale problem is well posed for specified Lévy-type operators.

Includes jump kernels of geometric stable processes with stability index α in (0,2].

Provides conditions under which the martingale problem is well defined.

Abstract

We prove that the martingale problem is well posed for pure-jump L\'evy-type operators of the form $$ (\mathcal Lf)(x) = \int_{\mathbb R^d \setminus \{0\}} \left(f(x+h)-f(x) - (\nabla f(x) \cdot h)1_{\|h\| < 1}\right)K(x,h) dh, $$ where $K(x,\cdot)$ is a jump kernel of the form $K(x,h) \sim \frac{l(\|h\|)}{\|h\|^d}$ for each $x \in \mathbb R^d,\|h\|<1$, and $l$ is a positive function that is slowly varying at $0$, under suitable assumptions on $K$. This includes jump kernels such as those of $\alpha$-geometric stable processes, $\alpha \in (0,2]$.