Chapter ownership and proof scope
Primary proof chapter: Chapter 31
Related tools and applications: Chapter 26 · Chapter 32 (not yet written)
- Written proof or calculation (limited scope)
A written quadratic-variation proof is available for fixed T along dyadic grids, assuming Brownian motion exists.
- Mathematical intuition
Fixed-time difference-quotient scaling motivates nondifferentiability; it is not a complete almost-sure nowhere-differentiability proof.
- Theorem still to be proved
Chapter 26 handles path regularity; Chapter 31 develops general variation and integration, and Chapter 32 proves Itô’s formula.
- Written proof or calculation (limited scope)
Chapter 26 adds a uniform-mesh nowhere-differentiability proof and deterministic dyadic square sums; general brackets and stochastic integration remain for Chapter 31.
- Written proof or calculation (limited scope)
Chapter31 and T-ITO-BRACKET contain complete working proofs for continuous-local-martingale integration, brackets and ucp square sums; author review and human final review remain open; the legacy body is preserved.
Tools needed by the individual arguments
Existence of Brownian motion, L2 convergence, Borel–Cantelli, stochastic integration, and Itô’s formula.
These states describe the existing material; they do not certify a complete theorem treatment or final mathematical review.
Advanced Reading: Path Roughness, Quadratic Variation, and the Motivation for Itô Calculus
4.10 Continuous Yet Not Differentiable: Another Paradox of Brownian Motion
We have now actually constructed Brownian motion.
It has continuous sample paths:
t\longmapsto B_t(\omega).At first glance, since the particle’s position changes continuously with time, it seems we should be able to define its instantaneous velocity:
v(t)=\lim_{h\to0}\frac{B_{t+h}-B_t}{h}.But matters are not that simple.
By the definition of Brownian motion,
B_{t+h}-B_t\sim N(0,h)for h>0.
Thus
\frac{B_{t+h}-B_t}{h}
\sim N\left(0,\frac1h\right).As h\downarrow0, the variance of this difference quotient actually tends to infinity.
The difference quotient therefore does not become increasingly concentrated near any finite value.
Indeed, for any fixed M>0,
P\left(
\left|\frac{B_{t+h}-B_t}{h}\right|\le M
\right)\longrightarrow0.These difference quotients cannot therefore even converge in probability to a finite random variable.
But this is still not the strongest conclusion.
Brownian sample paths are in fact almost surely nowhere differentiable.
This is a surprising fact:
\boxed{
\text{sample paths are continuous}
\quad\text{but almost surely nowhere differentiable}.
}In calculus, we became accustomed to an intuition:
Continuous functions represent smooth motion, and velocity should be obtained by differentiating position.
Brownian motion overturns this intuition completely.
It is continuous motion with a rigorous mathematical existence, yet its paths are too rough to describe using ordinary differentiation. For proofs and the historical development of path regularity, see modern probability texts on Brownian motion.〔45, 46〕
We can, however, understand this roughness through a more structural quantity.
4.10.1 How Rough Is a Path?
On the interval [0,T], consider the dyadic grid
t_k=\frac{kT}{2^m},
\qquad k=0,\ldots,2^m.Define the sum of squared increments:
Q_m=
\sum_{k=0}^{2^m-1}
(B_{t_{k+1}}-B_{t_k})^2.Since the increments are independent and
B_{t_{k+1}}-B_{t_k}
\sim N\left(0,\frac{T}{2^m}\right),we have
\mathbb E[Q_m]
=
2^m\cdot\frac{T}{2^m}
=T.A normal random variable with mean 0 and variance v, denoted by Z, satisfies
\operatorname{Var}(Z^2)=2v^2.Therefore
\begin{aligned}
\operatorname{Var}(Q_m)
&=
2^m\cdot2\left(\frac{T}{2^m}\right)^2\\
&=\frac{2T^2}{2^m}.
\end{aligned}Thus
Q_m\longrightarrow Tin L^2.
Furthermore, Chebyshev’s inequality gives
P(|Q_m-T|>\varepsilon)
\le
\frac{2T^2}{\varepsilon^2 2^m}.The right-hand side is summable in m.
The Borel–Cantelli lemma therefore yields
\boxed{
Q_m\xrightarrow{\mathrm{a.s.}}T.
}This is a rigorous statement of Brownian motion’s quadratic variation along dyadic grids.
We usually write
\boxed{[B]_T=T.}4.10.2 Why Is This Completely Different from Ordinary Smooth Functions?
Suppose f:[0,T]\to\mathbb R is continuously differentiable.
The mean value theorem gives
|f(t_{k+1})-f(t_k)|
\le M(t_{k+1}-t_k)where
M=\sup_{0\le t\le T}|f'(t)|<\infty.Hence
\begin{aligned}
\sum_k(f(t_{k+1})-f(t_k))^2
&\le
M^2\sum_k(t_{k+1}-t_k)^2\\
&=M^2\frac{T^2}{2^m}\\
&\longrightarrow0.
\end{aligned}For a smooth function, the sum of squared increments along increasingly fine grids therefore tends to zero.
Brownian motion, however, satisfies
\sum_k(B_{t_{k+1}}-B_{t_k})^2
\longrightarrow T.In other words:
\boxed{
\begin{aligned}
\text{smooth paths：}&\quad [f]_T=0,\\
\text{Brownian paths：}&\quad [B]_T=T.
\end{aligned}
}This goes deeper than saying that “the paths are somewhat unsmooth.”
It shows that Brownian motion has a cumulative second-order effect that ordinary calculus cannot disregard.
4.10.3 Why Must Stochastic Calculus Be Reinvented?
In ordinary calculus, we often disregard second-order infinitesimals.
Indeed, if
dx=O(dt),then
(dx)^2=O(dt^2),which tends to zero after division by dt.
For Brownian motion, however, typical increments have the scale
dB\sim O(\sqrt{dt})shown above.
Thus
(dB)^2\sim O(dt).The accumulated contribution of squared increments is therefore of the same order as the first-order time increments.
We can no longer disregard it as in ordinary calculus.
This is the fundamental reason why the formal relationship
\boxed{(dB_t)^2=dt}will appear in Itô calculus.
This is not algebra that treats dB_t as an ordinary real differential; it is concise notation for the quadratic-variation structure within the theory of stochastic integration.
For example, ordinary calculus might lead us to expect
d(B_t^2)=2B_t\,dB_t.whereas Itô’s formula actually gives
\boxed{
d(B_t^2)=2B_t\,dB_t+dt.
}The additional dt arises precisely from Brownian motion’s nonzero quadratic variation.
We have not yet established stochastic integration rigorously, so this remains a preview of the theory to come.
But it already reveals a very important pattern in the development of probability:
New theories of integration arise because new random objects cannot be fully described by existing calculus, rather than simply because mathematicians want more complicated formulas.