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, $\alpha \in (0,2]$.
Categories: math.PR