Higher-order derivatives of self-intersection local time for linear fractional stable processes

Abstract

In this paper, we aim to consider the $k=(k_1,k_2,\cdots,k_d)$-th order derivatives $\beta^{(k)}(T,x)$ of self-intersection local time $\beta(T,x)$ for the linear fractional stable process $X^{\alpha,H}$ in $\mathbb{R}^{d}$ with indices $\alpha\in(0,2)$ and $H=(H_{1},\cdots,H_{d})\in(0,1)^d$. We first give sufficient condition for the existence and joint H\"{o}lder continuity of the derivatives $\beta^{(k)}(T,x)$ using the local nondeterminism of linear fractional stable processes. As a related problem, we also study the power variation of $\beta^{(k)}(T,x)$.