Chapter 02

Medium

Brownian Motion: Definition and Path Properties

00 · Symbol Glossary

$W_t$W sub t — Wiener process / Brownian motion

The continuous-time limit built informally in Chapter 01, now given a precise axiomatic definition. "Wiener process" and "Brownian motion" are the same object; this text uses them interchangeably.

$W_t - W_s$increment over $[s,t]$

The change in the process between two times $s<t$ . Under the axioms below, this is normally distributed with mean $0$ and variance $t-s$ , and independent of everything that happened before time $s$ .

$N(0,t-s)$Normal with mean 0, variance $t-s$

The distribution of $W_t-W_s$ . Note the variance is the elapsed time itself, not a free parameter — Brownian motion has no adjustable variance rate unless one is added explicitly, e.g. $\sigma W_t$ .

$c^{-1/2}W_{ct}$rescaled path

The path of $W$ sped up or slowed down by a factor $c>0$ and rescaled in space by $\sqrt{c}$ . Section 04 shows this has exactly the same law as $W_t$ itself — Brownian motion looks statistically identical at every time scale.

01 · Defining the Wiener Process

Chapter 01 built a continuous-time limit out of a rescaled random walk and described what it should look like. This chapter turns that description into a definition that stands on its own, with no reference to a discrete walk underneath it.

Definition — Standard Wiener Process

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

\text{(i) } W_0 = 0 \text{ a.s.} \qquad \text{(ii) independent increments} \qquad \text{(iii) } W_t - W_s \sim N(0,\,t-s) \text{ for } t>s \qquad \text{(iv) } t\mapsto W_t \text{ is a.s. continuous}

Condition (ii) means that for any $0\le t_0<t_1<\cdots<t_n$ , the increments $W_{t_1}-W_{t_0},\ldots,W_{t_n}-W_{t_{n-1}}$ are mutually independent random variables — not merely uncorrelated, and not just pairwise independent of the immediately preceding increment.

Plain language: the process starts at zero, moves by increments that are unpredictable Gaussian noise scaled to the length of time elapsed, those increments never share information across non-overlapping time windows, and the path never jumps — it can be drawn without lifting the pen. A process built from a coefficient $\sigma$ on top of this, $\sigma W_t$ , has increment variance $\sigma^2(t-s)$ and plays the role of $X_t^{(n)}$ 's limit from Chapter 01.

02 · Why These Axioms, Not Others

Each axiom earns its place by ruling out a specific failure the discrete construction warns about.

Example — Where each axiom comes from

Independent increments (ii) is inherited directly from the i.i.d. steps $\xi_i$ of the discrete walk — nothing in the limiting operation introduces dependence across disjoint blocks of steps. The Gaussian shape in (iii) is the invariance principle from Chapter 01: any finite-variance step distribution converges to the same Gaussian limit by the Central Limit Theorem, regardless of the original step's shape. The variance $t-s$ (rather than some other function of $t-s$ ) is the $\sigma^2 t$ scaling forced by the unique $\sqrt{\Delta t}$ step size derived in Chapter 01, Section 02.

Condition (iv), continuity, is not automatic. The first three axioms only pin down the finite-dimensional distributions of the process — the joint law of $(W_{t_1},\ldots,W_{t_n})$ for any finite collection of times. Two processes can share every finite-dimensional distribution while one has continuous paths and the other does not.

❌ Assuming continuity is implied by the other three axioms

Take a process satisfying (i)–(iii) and, at one fixed but random time $\tau$ (say uniform on $[0,1]$ ), redefine $\tilde W_\tau = W_\tau + 1$ while leaving every other time unchanged.

Why it breaks: because $\{\tau\}$ has probability zero of colliding with any fixed time $t$ , this modification changes no finite-dimensional distribution — $\tilde W$ satisfies (i)–(iii) exactly as well as $W$ does. But $\tilde W$ has a discontinuity at $t=\tau$ with probability $1$ .

Consequence: (i)–(iii) alone specify a family of processes, only some of which have continuous paths. Continuity has to be imposed as its own axiom, or established via a separate existence theorem (Kolmogorov's continuity criterion) that selects the continuous representative. This text takes continuity as given by axiom (iv) and does not reprove that existence theorem.

03 · Continuous, Yet Nowhere Differentiable

Brownian motion is continuous by axiom. It is also, with probability $1$ , differentiable at no point at all — a fact that looks contradictory until the variance scaling is examined directly.

Theorem — Nowhere Differentiability (Statement)

With probability $1$ , the function $t\mapsto W_t$ has no point $t$ at which the derivative

\lim_{h\to 0}\frac{W_{t+h}-W_t}{h}

exists (as a finite real number).

The intuition follows directly from Chapter 01's scaling argument. $W_{t+h}-W_t\sim N(0,h)$ , so a typical size for the increment is $\sqrt{h}$ (one standard deviation). The difference quotient therefore has typical size $\sqrt{h}/h = 1/\sqrt h$ , which diverges as $h\to 0$ instead of converging to a finite slope. A rigorous proof strengthens this into an almost-sure statement holding simultaneously at every $t$ , but the scaling mismatch between $\sqrt h$ and $h$ is the entire mechanism.

❌ Writing $dW_t/dt$ as an ordinary derivative

A modeler differentiates $W_t$ term by term inside a formula, treating $dW_t/dt$ as a well-defined function of $t$ the way $dt^2/dt = 2t$ is.

Why it breaks: the limit defining that derivative does not exist at any $t$ , by the theorem above. There is no function to differentiate into.

Consequence: any calculation that manipulates $dW_t$ as if it were $f'(t)\,dt$ for a differentiable $f$ is not merely imprecise — it references an object that provably does not exist. Chapter 05 introduces the Itô integral specifically to give $dW_t$ a rigorous meaning as an integrator, without ever requiring a derivative $dW_t/dt$ to exist.

04 · Self-Similarity and the Markov Property

Two structural symmetries make Brownian motion tractable despite the roughness of Section 03.

Definition — Brownian Scaling (Self-Similarity)

For any constant $c>0$ , the rescaled process $\tilde W_t = c^{-1/2}W_{ct}$ is itself a standard Wiener process:

\{c^{-1/2}W_{ct}\}_{t\geq 0} \stackrel{d}{=} \{W_t\}_{t\geq 0}

Plain language: zoom in on any window of a Brownian path, rescale time and space by matching factors ( $c$ in time, $\sqrt c$ in space), and the rescaled picture is statistically indistinguishable from the original. There is no characteristic time scale at which the path "smooths out" — it looks equally jagged no matter how far you zoom in, which is consistent with nowhere-differentiability rather than contradicting it.

Markov, stated informally

Brownian motion is a Markov process: given the value $W_t$ at the present time, the distribution of the future $\{W_{t+h}\}_{h\geq0}$ does not depend on the path's history before $t$ . This follows directly from independent increments — $W_{t+h}-W_t$ is independent of $\{W_s\}_{s\le t}$ by construction, so the past enters the future only through the current value $W_t$ . Restarting the process at any time $t$ and shifting to $\tilde W_h = W_{t+h}-W_t$ produces a fresh standard Wiener process, independent of the path up to $t$ — the same restart property the discrete walk had in Chapter 01, Section 04.

05 · Exercises

EXERCISE 2.1

Use axiom (iii) directly with $s=2$ , $t=5$ .

$W_5-W_2 \sim N(0, 5-2) = N(0,3)$ . So $\mathbb{E}[W_5-W_2]=0$ and $\mathrm{Var}(W_5-W_2)=3$ ; the standard deviation is $\sqrt3\approx1.73$ .

For a standard Wiener process, state the exact distribution of $W_5 - W_2$ , including both parameters, citing the axiom used.

EXERCISE 2.2

Write $\mathrm{Cov}(W_s,W_t)$ for $s<t$ as $\mathrm{Cov}(W_s, W_s+(W_t-W_s))$ and expand using independent increments.

For $s<t$ : $\mathrm{Cov}(W_s,W_t) = \mathrm{Cov}(W_s,\,W_s+(W_t-W_s)) = \mathrm{Var}(W_s) + \mathrm{Cov}(W_s,\,W_t-W_s)$ . Since $W_s = W_s - W_0$ is an increment over $[0,s]$ and $W_t-W_s$ is the increment over $[s,t]$ , independent increments makes the second term $0$ . So $\mathrm{Cov}(W_s,W_t) = \mathrm{Var}(W_s) = s$ . In general, $\mathrm{Cov}(W_s,W_t) = \min(s,t)$ .

Derive $\mathrm{Cov}(W_s,W_t)$ for $s<t$ from the four axioms, and state the general formula for arbitrary $s,t\geq 0$ .

EXERCISE 2.3

Apply the scaling property with $c=4$ and compare $\mathrm{Var}(W_{4t})$ to $\mathrm{Var}(2W_t)$ .

Brownian scaling gives $W_{4t} \stackrel{d}{=} 2W_t$ (taking $c=4$ , so $c^{-1/2}=\tfrac12$ , i.e. $\tfrac12 W_{4t}\stackrel{d}{=}W_t$ , equivalently $W_{4t}\stackrel{d}{=}2W_t$ ). Checking variances: $\mathrm{Var}(W_{4t}) = 4t$ directly from axiom (iii), and $\mathrm{Var}(2W_t) = 4\,\mathrm{Var}(W_t) = 4t$ . Both sides match, confirming the scaling relation is consistent with the variance already known from the axioms — it is not new information about variance, but it does assert the entire distribution, not just variance, matches.

Use Brownian scaling with $c=4$ to relate $W_{4t}$ and $W_t$ in distribution, then verify the relation is consistent with the variance formula from axiom (iii).

06 · Chapter Summary

Concept	Meaning
Axioms (i)–(iv)	$W_0=0$ , independent increments, $W_t-W_s\sim N(0,t-s)$ , continuous paths
Continuity	not implied by (i)–(iii); must be imposed or proven via a separate theorem
Nowhere differentiable	difference quotient $\sim 1/\sqrt h \to \infty$ , so no slope exists anywhere
Self-similarity	$c^{-1/2}W_{ct}\stackrel{d}{=}W_t$ — no characteristic time scale
Markov property	future given $W_t$ is independent of the path before $t$
$\mathrm{Cov}(W_s,W_t)$	$\min(s,t)$

Next: Chapter 03 — Quadratic Variation and Why Calculus Breaks, which turns the roughness described here into a precise quantity — $[W,W]_t=t$ — and shows exactly where the Riemann–Stieltjes integral fails on Brownian paths.