Let $X$ be a continuous semimartingale and $L^x_t$ be its local time on the interval $[0,t]$ at level $x$. Setting $s \geq 1$, I was wondering if any bounds were known regarding the growth of $\mathbb{E}\left[(L_t^x)^s\right]$ as a function of $t$ ? Intuitively, I would like to say that, as a function of $t$, this can be bounded by for instance in terms of the quadratic variation of the process $[X]_t$, but haven't really been able to find results in this direction.
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