Brownian Motion and Quadratic Variation

Setup

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a complete probability space equipped with a filtration $\mathbb{F} = (\mathcal{F}_t)_{t \geq 0}$ satisfying the usual conditions: right-continuity ($\mathcal{F}_t = \mathcal{F}_{t+}$) and completeness (null sets of $\mathbb{P}$ are in $\mathcal{F}_0$).

All processes are defined on this space. Time horizon is $[0, T]$ for a fixed $T > 0$. Expectations are under $\mathbb{P}$ unless stated otherwise.

Definition

A standard Brownian motion (or Wiener process) is a stochastic process $(W_t)_{t \geq 0}$ satisfying:

1. Initial condition: $W_0 = 0$ almost surely.
2. Independent increments: for $0 \leq s < t$, the increment $W_t - W_s$ is independent of $\mathcal{F}_s$.
3. Gaussian increments: $W_t - W_s \sim \mathcal{N}(0,\, t - s)$ for all $0 \leq s \leq t$.
4. Continuous paths: $t \mapsto W_t(\omega)$ is continuous $\mathbb{P}$-almost surely.

These axioms are not redundant. Axiom 4 is a topological regularity condition, independent of the distributional structure in axioms 2–3. Without it, the process would be defined only up to modification, and pathwise integration would be meaningless.

The covariance structure follows immediately: for $s \leq t$, $\mathrm{Cov}(W_s, W_t) = \mathbb{E}[W_s W_t] = s.$

Sample Path Properties

Hölder Continuity

By the Kolmogorov–Chentsov continuity theorem, a process with $\mathbb{E}[|W_t - W_s|^p] \leq C |t - s|^{1 + \alpha}$ for some $p, \alpha, C > 0$ admits a version with Hölder-continuous paths of any exponent $\gamma < \alpha/p$.

For Brownian motion, $\mathbb{E}[|W_t - W_s|^{2k}] = (2k-1)!! \cdot |t-s|^k$. Taking $k = 2$: $\mathbb{E}[|W_t - W_s|^4] = 3(t-s)^2$, so we may take $p = 4$, $\alpha = 1$, giving Hölder exponent up to $\gamma < 1/4$. Sharper analysis yields Hölder exponent $\gamma < 1/2$.

Brownian motion paths are not Hölder-$1/2$: the modulus of continuity is $\lim_{h \to 0} \sup_{|t - s| \leq h} \frac{|W_t - W_s|}{\sqrt{2h \log(1/h)}} = 1 \quad \text{a.s.}$ (the Lévy modulus). The $\sqrt{h \log(1/h)}$ factor, not just $\sqrt{h}$, captures the exact regularity.

Nowhere Differentiability

With probability one, Brownian motion is nowhere differentiable. To see why: if the path were differentiable at some point $t$, then $(W_{t+h} - W_t)/h \to W'_t$ as $h \to 0$. But this quotient has standard deviation $h^{-1/2}$, which diverges. A formal proof uses Paley–Wiener–Zygmund or a direct Borel–Cantelli argument.

This is the mathematical source of the informal statement "Brownian motion fluctuates on every scale."

Infinite Total Variation

Define the total variation of $W$ on $[0, T]$ over a partition $\pi = \{0 = t_0 < t_1 < \cdots < t_n = T\}$: $TV(W, \pi) = \sum_{i=1}^{n} |W_{t_i} - W_{t_{i-1}}|.$

The total variation of Brownian motion is infinite almost surely: $\sup_{\pi} TV(W, \pi) = +\infty \quad \text{a.s.}$

This has a fundamental consequence: Lebesgue–Stieltjes integration of the form $\int f \, dW$ cannot be defined pathwise. The standard integration-by-parts formula requires finite variation. For Brownian motion, a new theory — the Itô integral — is required.

Quadratic Variation

The quadratic variation of a process $X$ on $[0, t]$ is defined as the limit in probability: $[X]_t = \lim_{|\pi| \to 0} \sum_{i=1}^{n} (X_{t_i} - X_{t_{i-1}})^2,$ where the limit is taken as the mesh $|\pi| = \max_i(t_i - t_{i-1}) \to 0$ over arbitrary partitions.

Theorem: $[W]_t = t$

For standard Brownian motion, $[W]_t = t$ almost surely for all $t \geq 0$.

Proof. Fix an equal-mesh partition with $n$ intervals and mesh $h = t/n$. Let $\Delta_i = W_{t_i} - W_{t_{i-1}}$ and define: $Q_n = \sum_{i=1}^{n} \Delta_i^2.$

Since $\Delta_i \sim \mathcal{N}(0, h)$ and all $\Delta_i$ are independent: $\mathbb{E}[\Delta_i^2] = h, \qquad \mathrm{Var}(\Delta_i^2) = \mathbb{E}[\Delta_i^4] - h^2 = 3h^2 - h^2 = 2h^2.$

Therefore: $\mathbb{E}[Q_n] = nh = t, \qquad \mathrm{Var}(Q_n) = n \cdot 2h^2 = 2t^2/n \xrightarrow[n \to \infty]{} 0.$

By Chebyshev's inequality, $Q_n \to t$ in $L^2$, and hence in probability. Convergence a.s. follows by a subsequence argument extended to general (non-equal) partitions via an $L^2$ monotonicity argument. $\square$

Differential Notation

The result is commonly written as: $(dW_t)^2 = dt.$

This is shorthand for the quadratic variation result, not a pathwise identity. More precisely, for any adapted process $f$, $\int_0^T f_t \, d[W]_t = \int_0^T f_t \, dt.$ This identity is central to the derivation of Itô's lemma.

Contrast: Total Variation vs Quadratic Variation

For a $C^1$ function $f$ on $[0,T]$: increments are $O(h)$, so squared increments are $O(h^2)$, and the sum over $n = T/h$ partitions is $O(h) \to 0$.

For Brownian motion: increments are $O(\sqrt{h})$, squared increments are $O(h)$, and the sum over $n = T/h$ partitions converges to $T$. The non-trivial quadratic variation is the source of the Itô correction in the stochastic chain rule.

Martingale Property

Brownian motion is a martingale with respect to its natural filtration $\mathcal{F}^W_t = \sigma(W_s : s \leq t)$: $\mathbb{E}[W_t \mid \mathcal{F}_s] = W_s, \quad s \leq t.$

This follows immediately from independent increments: $\mathbb{E}[W_t - W_s \mid \mathcal{F}_s] = \mathbb{E}[W_t - W_s] = 0$.

Moreover, $W_t^2 - t$ is a martingale: $\mathbb{E}[W_t^2 - t \mid \mathcal{F}_s] = W_s^2 - s.$

The subtracted compensator $t$ is precisely the quadratic variation $[W]_t$. In general, for any continuous local martingale $M$, the process $M_t^2 - [M]_t$ is a local martingale. This is the Doob–Meyer decomposition in the continuous case.

Lévy's Characterisation

Theorem (Lévy). Let $M = (M_t)_{t \geq 0}$ be a continuous local martingale with $M_0 = 0$ and $[M]_t = t$ for all $t \geq 0$. Then $M$ is a standard Brownian motion.

This is used in proving Girsanov's theorem: after a change of measure, Lévy's characterisation identifies the new process as a Brownian motion by verifying continuity, the local martingale property, and the quadratic variation.

Limitations

Continuous-time idealization. Brownian motion is a mathematical model, not an empirical law. Real asset prices trade discretely, are bounded below by zero, and exhibit microstructure noise at fine timescales. The GBM assumption is a modelling choice.

No-jump model. Standard Brownian motion has continuous paths. Models incorporating jumps (Poisson processes, Lévy processes) require a separate theory; the Itô calculus developed from Brownian motion does not directly apply.

Fractional Brownian motion. For $H \neq 1/2$, fractional Brownian motion (fBM) has correlated increments and, for $H \neq 1/2$, is not a semimartingale. The standard Itô calculus is invalid for fBM, requiring rough path theory or Malliavin calculus. Rough volatility models (Gatheral–Jaisson–Rosenbaum, 2018) use fBM with $H \approx 0.1$ to model instantaneous variance.