brownian motion unbounded variation I have been doing a little bit of reading regarding random processes and probability theory recently for some research I have been doing, and I have come across the claim in many places that Brownian motion cannot be treated with Riemannian integration due to the fact that it is of unbounded variation. I have been trying to find a rigorous proof for that, but I have been having a difficult time. I intuitively have the idea that since Brownian motion is considered as a continuous random walk, then it is theoretically possible for it to exceed whatever bound may be placed on it. Is this the right way of thinking of it? And can anyone produce  more rigorous proof to show this? Thanks!
 A: One way to rigorously formulate what you are talking about is as follows. As a process, the sample paths of the usual Brownian motion $(B_t)_{t\in\mathbb R_+}$ are not functions of bounded variation. In fact, almost surely, the sample paths have infinite variation on any interval of the form $[0,t]$ with $t\in\mathbb R_+$.
This presents the following issue if we want to try to define what $\int_0^t X_s\,dB_s$ means, where $(X_t)_{t\in\mathbb R_+}$ is a suitable random process. In the Riemann-Stieltjes approach to integration, we can define integrals of the form $\int_0^t f(s)\,d\alpha(s)$ when $\alpha$ is, say, a function of bounded variation on $[0,t]$, and $f$ is a continuous function on $[0,t]$. When $\alpha$ does not have bounded variation, then there are continuous functions $f$ that are not Riemann-Stieltjes integrable with respect to $\alpha$.
So this means we have to look for another way to interpret what $\int_0^tX_s\,dB_s$ should mean, and the way it is treated in Le Gall's book Brownian Motion, Martingales, and Stochastic Calculus,
for instance, is to proceed by building up the theory of martingales and stochastic integration enough so that we can define expressions of the form $\int_0^t X_s\,dB_s$ as stochastic processes known as stochastic integrals, which are very roughly speaking some kind of martingale-like object.
Le Gall's book is a great reference for this and other topics related to Brownian motion after one has a suitable background in probability theory.
A: The book "Brownian motion and stochastic calculus" by Karatzas and Shreve has your answer.  The proof is a bit scattered but proceeds something like this:
Section 1.5 shows that if the p^th variation of a continuous square integrable martingale converges in probability to some process that q^th variation for q< p is infinity.  Also q^th variation for q>p is 0.
Next, show that Brownian motion has quadratic variation t.  (somewhere in chapter 2 probably). Since Brownian motion is continuous square integrable martingale, this implies that first variation is infinite.
Pretty technical stuff.  Requires several lemmas and other results.
