Brownian Motion and Continuity

Question

Consider a Brownian Motion $(B_{t})_{t\geq0}$. In my lecure notes it says, without proof, that $\mathbb{P}\left(\sup_{t,s\leq N}\left\{ \left|B_{t}-B_{s}\right|:\left|t-s\right|<\delta\right\} <\varepsilon\right)$ converges to $0$ for $\delta$ tending to $0$. I think it is a consequence of the continuity of the sample paths, but can anyone help with a proof?

Answer 1

First there's a typo: you should have $>\varepsilon$ or should say that the probability converges to to $1$, not $0$.

Anyway, here's my proof. It's informal and skips over details; you should fill these in.

Fix $t$ and $s$ for a moment. Then your probability is the probability that a normal variable centered at zero whose variance is at most $\delta$ is outside the interval $(-\varepsilon,\varepsilon)$. This goes to zero by Markov's inequality (or Chebyshev's inequality, different authors seem to use different names for this).

Now fix $s$ and let $t \geq s$ be arbitrary. Then the Markov property tells us that $B_t - B_s$ is another Brownian motion, call it $B'_{t-s}$. Doob's maximal inequality estimates your probability by a constant times the corresponding probability for $B'_{N-s}$, which goes to zero by the previous argument.

Now allow $s$ to vary as well. Then the probability still goes to zero, because the suprema over $t$ from the previous paragraph are bounded by our estimate for the supremum over $t$ with $s=0$.

Answer 2

Another approach would be to use the continuity of the sample paths directly. As above, I'm assuming you mean that the probability converges to $1$ rather than $0$.

Using the density of the rationals in the reals, and then upwards continuity of the probability measure, we get:

$\lim_{\delta\rightarrow0}\mathbb{P}\left(\sup_{t,s\leq N}\left\{ \left|B_{t}-B_{s}\right|:\left|t-s\right|<\delta\right\} <\varepsilon\right) = \lim_{\delta\rightarrow0}\mathbb{P}\left(\bigcap_{t,s\in\left[0,N\right]}\left\{ \left|B_{t}-B_{s}\right|<\varepsilon:\left|t-s\right|<\delta\right\} \right) = \lim_{n\rightarrow\infty}\mathbb{P}\left(\bigcap_{t,s\in\left[0,N\right]\cap\mathbb{Q}}\left\{ \left|B_{t}-B_{s}\right|<\varepsilon:\left|t-s\right|<\frac{1}{n}\right\} \right) = \mathbb{P}\left(\bigcup_{n\geq0}\bigcap_{t,s\in\left[0,N\right]\cap\mathbb{Q}}\left\{ \left|B_{t}-B_{s}\right|<\varepsilon:\left|t-s\right|<\frac{1}{n}\right\} \right) = \mathbb{P}\left(\exists\delta>0\;\mbox{s.t.}\;\left|B_{t}-B_{s}\right|<\varepsilon\;\mbox{whenever}\;\left|t-s\right|<\delta\;\mbox{and}\; t,s\in[0,N]\right)$

The latter probability is $1$ since $t\mapsto B_{t}\left(\omega\right)$ is uniformly continuous on $[0,N]$.