On the "uniform continuity" of Brownian motion under expectation

Question

Let $(W_t)_{t\ge 0}$ be a standard Brownian motion. For each $t\in [0,1]$, it is known that, e.g. from Burkholder-Davis-Gundy's inequality

$$\mathbb E\big[\sup_{s\in [t,t+\Delta t]}|W_s-W_t|^p\big]=O(\Delta t^{p/2}),\quad \forall p\ge 1,$$

where $O$ refers to "of order". Do we have an estimate of

$$\mathbb E\big[\sup_{s,t\in[0,1],~ |s-t|\le\Delta t}|W_s-W_t|^p\big]?$$

A student asked me this question when I taught Euler's scheme applied to SDEs and its convergence, but I cannot find any reference. Any answers, comments or references are appreciated.

Answer 1

For $n\in\mathbb{Z}_{\geq 0}$ and $0\leq i< 2^n$, denote
$$ X_{n,i}=\sup_{t\in[i2^{-n}, (i+1)2^{-n}]}{|W_t-W_{i2^n}|}. $$

Let $n$ be such that $2^{-{n}}<|\Delta t|\leq 2^{-{n+1}}$. Then, $$\sup_{s,t\in[0,1],~ |s-t|\le\Delta t}|W_s-W_t|^p\leq 4\sup_{i}|X_{i,n}|^p,$$ because such $s$ and $t$ must belong to a union of three consecutive diadic intervals of length $2^{-n}$. For a fixed $n$, $X_{n,i}$ are i. i. d. random variables; moreover, by Brownian scaling, $$ X_{n,i}\stackrel{\mathcal{D}}{=}2^{-n/2}X_{0,0}. $$ If $F(t)$ denotes the cumulative disctribution function of $X_{0,0}$, then we have, with $N=2^n,$ $$\mathbb{P}(\sup_{i}|X_{i,n}|\leq t)=F(N^{1/2} t)^{N}.$$ It is well known that $\mathbb{P}(\sup_{[0,1]} W_t>a)=2\mathbb{P}(W_t>a)\leq 2e^{-a^2/2}$ for $a$ large enough, so that $F(t)\geq 1-4e^{-t^2/2}$, and $$ \mathbb{P}(\sup_{i}|X_{i,n}|>t)\leq 1-(1-4e^{-Nt^2/2})^{N}. $$ Let $T_n:=\sqrt{2\log 2}n^\frac12 N^{-\frac12}$, then, plugging into the above, $$ \mathbb{P}(\sup_{i}|X_{i,n}|>kT_n)\leq 1-e^{N\log(1-4\cdot 2^{-k^2n})}\leq 10\cdot 2^{(1-k^2)n}. $$ That is to say, the median of $\sup_{i}|X_{i,n}|$ is of order at most $T_n$ and its tail decays very sharply after that, implying
$$\mathbb{E}\sup_{i}|X_{i,n}|^p=O(T_n^p)$$ for all $p$, or $$\mathbb{E}\left(\sup_{s,t\in[0,1],~ |s-t|\le\Delta t}|W_s-W_t|^p\right)=O\left((|\Delta t\log (\Delta t)|^\frac{p}{2}\right).$$

These estimates don't lose much and they are of course very similar to Lévy's theorem on the modulus of continuity for Brownian motion.