So I'm given a BM till time 't', and I'm asked what is the length of the path. As BM is not differentiable, I just can't directly use arc of length results for calculus. My intuition is each fraction of the path looks exactly like the bigger path of time t. So in a way even if consider a infinitesimal small interval I still have the same jagged BM, so the length of the path doesn't make sense directly to me?
Can someone please quantify this, and better explain the intution.
Please let me know if something is not clear, I'll try to improve my explanation.
A curve induced by $f:[a,b]\to\mathbb{R}$ has finite length iff $f\in \text{BV}[a,b]$ (see, e.g., this question). However, a function of bounded variation on $\mathbb{R}$ must be differentiable almost everywhere. Consequently, since the sample paths of a BM are a.s. nowhere differentiable, these paths are a.s. non-rectifiable (i.e. have infinite length) over any time interval.