Let $(X\_t)\_{t \in \mathbb{N}}$ be a real-valued martingale that is bounded, i.e.,
there are $a, b \in \mathbb{R}$ such that $a \leq X\_t \leq b$ for all $t$.


Define the *path length* $L$ of $(X\_t)\_{t \in \mathbb{N}}$ as the distance the martingale covers,
$$
L := \sum\_{t=0}^\infty | X\_t - X\_{t+1} |.
$$


My questions:


1. Is $L$ almost surely finite? I.e., can there be a set of positive measure on which $L = \infty$?
2. If yes to question 1, is $\mathbb{E}[L]$ finite?



Bounded martingales of infinite path length
If you have a random walk on $\{0,\ldots,N\}$, it's known that the expected
number of steps to hit the boundary starting from the middle is something like $N^2/4$. That is: you travel a distance of $N^2/4$ before hitting the boundary. 


You can exploit this to produce a martingale where the expected path length is infinite: Do absorbing random walk on $[0,1]$, where the step sizes are dyadic 
and depend on time. For example, do $2^{10}$ steps of size $2^{-10}$; then $2^{11}$ steps of size $2^{-11}$ etc. In the $2^{-k}$ phase, your average displacement is something like $2^{-k/2}$, so that it's likely that you'll never hit the boundary.


But any sample path that doesn't hit the boundary has infinite length.


