Brownian Motion — Mathematics Knowledge System

Overview

Brownian motion (or the Wiener process) is the canonical continuous-time stochastic process with independent Gaussian increments. It is the scaling limit of random walks, serves as the driving noise in stochastic differential equations, and provides the model for diffusion, heat flow, and financial price movements. Despite almost surely being nowhere differentiable, Brownian paths are Hölder-continuous of any order less than 1/2 — a perfect instantiation of the fractal roughness of nature.

Intuition

Imagine zooming in on a Brownian path — no matter how much you zoom, the path looks equally irregular. This is self-similarity: B_{at} ~ √a · B_t in distribution. The path is continuous (no jumps) but has infinite variation over any interval, making ordinary calculus inapplicable and necessitating Itô calculus.

Formal Definition

Definition

A standard Brownian motion (Wiener process) is a stochastic process {B_t}_{t≥0} satisfying:

B_0 = 0 \quad \text{a.s.}

Starts at zero

BM-0

t \mapsto B_t(\omega) \text{ is continuous a.s.}

Almost sure path continuity

BM-1

B_t - B_s \sim \mathcal{N}(0, t-s) \quad \forall 0 \leq s < t

Gaussian increments

BM-2

B_{t_1}, B_{t_2}-B_{t_1}, \ldots, B_{t_n}-B_{t_{n-1}} \text{ independent for } 0 \leq t_1 < \cdots < t_n

Independent increments

BM-3

Notation

Notation	Meaning
$B_t$	Standard Brownian motion at time t
$W_t$	Alternative notation (Wiener process)
$B^\mu_t = \mu t + \sigma B_t$	Brownian motion with drift μ and volatility σ

Properties

Gaussian process

\text{Any finite collection } (B_{t_1}, \ldots, B_{t_n}) \text{ is jointly Gaussian with } \text{Cov}(B_s, B_t) = \min(s,t)

Condition: Characterises Brownian motion among Gaussian processes

Martingale

\{B_t, \mathcal{F}_t\} \text{ is a martingale}

Condition: Natural filtration F_t = σ(B_s: s≤t) augmented

Quadratic variation

\langle B \rangle_t = t \quad \text{a.s.}

Condition: Limit of Σ(B_{t_{k+1}} - B_{t_k})² over partitions as mesh → 0

Self-similarity

\{B_{at}\}_{t\geq 0} \overset{d}{=} \{\sqrt{a}\,B_t\}_{t \geq 0} \quad \forall a > 0

Condition: Scaling property

Time reversal

\{B_T - B_{T-t}\}_{0 \leq t \leq T} \overset{d}{=} \{B_t\}_{0 \leq t \leq T}

Nowhere differentiability

\text{With probability 1, } t \mapsto B_t \text{ is nowhere differentiable}

Condition: Despite being continuous; a consequence of infinite quadratic variation

Law of iterated logarithm

\limsup_{t \to \infty} \frac{B_t}{\sqrt{2t \log\log t}} = 1 \quad \text{a.s.}

Condition: Khinchin's LIL

Theorems

Theorem 1: Lévy's Characterisation of Brownian Motion

\text{A continuous local martingale } M_t \text{ with } M_0 = 0 \text{ and } \langle M \rangle_t = t \text{ is a standard Brownian motion}

Theorem 2: Donsker's Invariance Principle

\text{If } S_n = \sum_{k=1}^n X_k \text{ with i.i.d. } X_k, E[X_1]=0, E[X_1^2]=\sigma^2, \text{ then } \frac{S_{\lfloor nt \rfloor}}{\sigma\sqrt{n}} \xrightarrow{d} B_t \text{ in } C([0,1])

Theorem 3: Reflection Principle

P\!\left(\max_{0 \leq s \leq t} B_s \geq a\right) = 2P(B_t \geq a) \quad \text{for } a > 0

Applications

The Black–Scholes model drives asset prices by geometric Brownian motion dS_t = μS_t dt + σS_t dB_t.

Worked Examples

By the reflection principle, for a > 0:
$P(M_t \geq a) = P\!\left(\max_{0\leq s\leq t} B_s \geq a\right) = 2P(B_t \geq a)$
Since B_t ~ N(0,t):
$P(M_t \geq a) = 2\left(1 - \Phi\!\left(\frac{a}{\sqrt{t}}\right)\right) = 2\bar{\Phi}\!\left(\frac{a}{\sqrt{t}}\right)$
Differentiating to get the density:
$f_{M_t}(a) = \sqrt{\frac{2}{\pi t}}\, e^{-a^2/(2t)} \quad a > 0$

Answer: M_t = max B_s has the same distribution as |B_t|, i.e. a folded normal distribution.

Practice Problems

Difficulty 8/10

Prove that {B_t² - t} is a martingale with respect to the natural filtration of Brownian motion.

Difficulty 9/10

State and apply the law of the iterated logarithm to determine, almost surely, the lim sup of B_t/t as t → ∞.

Difficulty 9/10

Use Lévy's characterisation to prove that if B_t is a standard Brownian motion and T_a = inf{t: B_t = a}, then {B_{t+T_a} - a}_{t≥0} is again a standard Brownian motion, independent of F_{T_a}.

Common Mistakes

Common Mistake

Brownian paths have finite variation

Brownian paths have infinite total variation on any interval a.s. This is why ordinary Riemann–Stieltjes integration against dB_t does not work and Itô calculus is needed.

Common Mistake

The quadratic variation [B]_t = t is a random variable that fluctuates

The quadratic variation of Brownian motion is identically t (deterministic). This is what makes the Itô isometry exact: E[(∫H dB)²] = E[∫H² dt].

Historical Background

Robert Brown (1827) observed the erratic motion of pollen grains under a microscope. Einstein (1905) explained it as bombardment by water molecules. Wiener (1923) constructed the rigorous mathematical model — now called the Wiener process — proving existence via a measure on path space. Lévy (1948) studied its fine path properties in depth.

1827
Brown observes irregular motion of pollen grains
Robert Brown
1905
Einstein's kinetic theory explanation
Albert Einstein
1923
Wiener constructs the rigorous mathematical Brownian motion
Norbert Wiener
1944
Itô's stochastic integral built on Brownian motion
Kiyosi Itô
1948
Lévy's Processus Stochastiques et Mouvement Brownien
Paul Lévy

Summary

Brownian motion is the unique (in law) continuous process with B_0=0, independent Gaussian increments, and Cov(B_s,B_t) = min(s,t).
Key properties: martingale, quadratic variation [B]_t = t, nowhere differentiable paths, self-similar.
The reflection principle gives the distribution of the running maximum.
Donsker's theorem: scaled random walks converge weakly to Brownian motion.
Brownian motion is the building block for Itô calculus and stochastic differential equations.

References

BookMörters, P. & Peres, Y. (2010). Brownian Motion. Cambridge University Press.
BookKaratzas, I. & Shreve, S. (1991). Brownian Motion and Stochastic Calculus. Springer.