For a diffusion $X_t$, I can set $$[X]^N_t = \sum_{j=1}^N \bigl(X_{t\frac{j}{N}}-X_{t\frac{j-1}{N}}\bigr)^2$$ Then it is well-known that the process $[X]^N_t$ tends to the quadratic variation $[X]_t$ uniformly on compact sets in probability, as $N \longrightarrow \infty$.
Question: Do they also converge in $L^p$ for some or all $1 \leq p < \infty$, i.e. $$\mathbb{E}\Bigl[\int_0^t\bigl|[X]^N_s-[X]_s\bigr|^p\mathrm{d} s\Bigr] \longrightarrow 0$$ as $N \longrightarrow \infty$.
Example: For Brownian motion in $\mathbb{R}^n$ (where $[X]_t = t$) I calculated that $$\mathbb{E}\Bigl[\bigl|[X]^N_t-[X]_t\bigr|^2\Bigr] = 2t^2\frac{1}{N},$$ so that the answer is yes for $p=2$ in this case.
What about general diffusion processes and general exponents?
\Edit: Nate Eldredge remarks that the quadratic variation need not be $L^p$. I restricted the question to Diffusions, where the quadratic variation should be $L^p$ for any $p$ under mild boundedness conditions on the coefficients.
