Remark on locally constant self-similar processes

Let $X=\{X(t),\ t\in\mathbb R_+\}$ be a self-similar process with index $\alpha>0$. We show that if $X$ is locally constant, and if $\mathbf P\{X(1)=0\}=0$, then the law of $X(t)$ is absolutely continuous. The applications of this result to homogeneous functionals of a multi-dimensional fractional Brownian motion are discussed.