modulus of continuity of Ito process We know from Levy's (uniform) modulus of continuity that for Brownian Motion, almost surely any sample path is locally Holder continuous for any $\rho <\frac{1}{2}$, i.e.
$$ |W_t - W_s | \leq C(\omega) |t-s|^\rho$$
for some path-dependent constant $C(\omega)$. Now I'm wondering if there is a similar result for Ito process/semimartingale, which is of the form
$$ dX_t = \mu_tdt + \sigma_tdW_t,$$
suppose $\mu_t$ and $\sigma_t$ satisfies any integrability condition (but may be random).
I would like to emphasize here that I'm looking for the PATHWISE continuity instead of the moments.
 A: One answer to this question is given in the article "On the Moments of the Modulus of Continuity of Ito Processes", by M. Fischer and G. Nappa in Stochastic Analysis and Applications 2009, p. 103-122.
Denote the modulus of continuity of a function $f:[0,\infty) \to \mathbb{R}^d$ by
$$ w_f(h,T) := \sup_{t,s  \in [0,T], |t-s| \le h} |f(t)-f(s)|, \quad h \ge 0, T>0. $$
Consider a $d_1$-dimensional Brownian motion adapted to a filtration $(\mathcal{F}_t)_{t \ge 0}$ satisfying the usual conditions. 
Let $p \ge 1$. Theorem 1 in the article states that if $X$ is an $d$-dimensional Ito process of the form
$$ dX_t = \mu_t dt + \sigma_t dW_t $$
and if there are the following bounds on $\mu$ and $\sigma$
$$ \int_s^t |\mu_i(u,\omega)|du \le \zeta(\omega) \cdot \sqrt{|t-s|\log\Big(\frac{2T}{|t-s|}\Big)}, $$
$$ \int_s^t |\sigma_{ij}^2(u,\omega)|du \le \xi(\omega) \cdot |t-s|, $$
for $t,s \in [0,T]$ and $1 \le i \le d$, $1 \le j \le d_1$ and $\zeta$,$\xi$ being $\mathcal{F}_T$-measurable random variables with values in $[0,\infty]$ such that there is $\varepsilon > 0$ with
$$  E[\zeta^p] < \infty, $$
$$ E[\xi^{\tfrac{p}{2}+\varepsilon}] < \infty, $$
then there is a finite constant $C(p)$ with
$$ E\big[ (w_X(h,T)^p \big] \le C(p) \big( h \log \big(\tfrac{2T}{h}\big) \big)^{\tfrac p 2}, $$
for all $h \in (0,T]$.
