Phase 8 - Lesson 8.3

Brownian Motion: Construction and Defining Properties

The continuous-time limit of the random walk: its four defining axioms, existence, the Gaussian marginal structure, and why the paths are continuous.

⏱ 55 min● Advanced🔗 Prereqs: 8.2, 7.6

↖ Phase 8 hub

Builds on: 8.2 rescaled the random walk; 7.6 gave the CLT that produces its Gaussian marginals.
Leads to: Its quadratic variation (8.4) and Gaussian structure (8.5) underpin the Ito calculus of Phase 9.

Learning Objectives

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State the four defining properties of standard Brownian motion.
Explain how the rescaled random walk converges to Brownian motion (Donsker’s idea).
Compute means, variances, and covariances of Brownian increments.
Describe the sample-path regularity: continuity everywhere, differentiability nowhere.

Key Vocabulary

Brownian motion: A process \((B_t)_{t\ge0}\) with \(B_0=0\), independent stationary Gaussian increments, and continuous paths.

Independent increments: For \(t_0\lt \cdots\lt t_k\), the increments \(B_{t_{i}}-B_{t_{i-1}}\) are mutually independent.

Stationary increments: \(B_{t}-B_{s}\sim\Normal(0,t-s)\) depends only on the gap \(t-s\).

Covariance function: \(\Cov(B_s,B_t)=\min(s,t)\).

Donsker’s theorem: The rescaled random walk \(S_{\lfloor nt\rfloor}/\sqrt n\) converges in distribution to Brownian motion.

Wiener process: Synonym for standard Brownian motion; the measure it induces on path space is Wiener measure.

Intuition & Motivation

Intuition

Brownian motion is what the random walk becomes when you take infinitely many, infinitely small steps. Shrink each step to \(\pm\sqrt{\Delta t}\) and take \(\Delta t\to0\): the CLT turns each increment Gaussian, independence survives, and the jagged path becomes continuous. The result is the canonical continuous-time process - the diffusion at the heart of every option-pricing model. Its four axioms are deceptively simple, yet they force a startling paradox: the paths are continuous everywhere but differentiable nowhere. There is no velocity, only variance; you cannot ask ‘how fast’, only ‘how spread out’. That is precisely why ordinary calculus fails and Ito’s calculus is needed.

Definition and existence

Definition - Standard Brownian motion

A stochastic process \((B_t)_{t\ge0}\) is a standard Brownian motion if: (1) \(B_0=0\); (2) it has independent increments; (3) for \(s\lt t\), \(B_t-B_s\sim\Normal(0,t-s)\) (stationary, Gaussian); (4) \(t\mapsto B_t\) is continuous almost surely.

Existence is not obvious - one must build a process with all four properties simultaneously. Two standard constructions: the Levy-Ciesielski series using a Schauder/Haar basis (guaranteeing continuity), and Donsker’s invariance principle, which realizes BM as the scaling limit of the random walk.

Theorem - Donsker’s invariance principle (idea)

Let \(S_n\) be a random walk with mean-zero, unit-variance increments. The linearly interpolated, rescaled path

\[W^{(n)}_t=\frac{S_{\lfloor nt\rfloor}}{\sqrt n}\ \Longrightarrow\ B_t\quad(n\to\infty),\] (8.4)

converges in distribution (on path space) to Brownian motion. The CLT gives the Gaussian marginals; the diffusive \(\sqrt n\) scale gives the right variance growth.

Moment and covariance structure

From the axioms, for \(s\le t\):

\[\E[B_t]=0,\quad \Var(B_t)=t,\quad \Cov(B_s,B_t)=\min(s,t).\] (8.5)

The covariance follows by writing \(B_t=B_s+(B_t-B_s)\) and using independence: \(\Cov(B_s,B_t)=\Var(B_s)+\Cov(B_s,B_t-B_s)=s+0=s=\min(s,t)\). Because all finite collections \((B_{t_1},\dots,B_{t_k})\) are jointly Gaussian, this mean-zero, \(\min(s,t)\)-covariance structure characterizes BM completely (it is a Gaussian process - see 8.5).

Path regularity: continuous but nowhere differentiable

Property (4) makes the paths continuous. Yet the increment over \([t,t+h]\) has size of order \(\sqrt h\), so the difference quotient \((B_{t+h}-B_t)/h\) is of order \(1/\sqrt h\to\infty\). Formally:

Theorem - Non-differentiability (Paley-Wiener-Zygmund)

With probability 1, the Brownian path \(t\mapsto B_t\) is nowhere differentiable and has infinite variation on every interval.

This is the geometric origin of the whole theory: a path that wiggles so violently that it has no slope but does have a well-defined quadratic variation. The next lesson makes that precise.

Worked Example - Compute a conditional forecast and its variance

Question: given \(B_1=0.4\), what is the distribution of \(B_3\)?

Decompose \(B_3=B_1+(B_3-B_1)\). The increment \(B_3-B_1\sim\Normal(0,3-1)=\Normal(0,2)\) and is independent of \(B_1\).

So conditional on \(B_1=0.4\): \(B_3\sim\Normal(0.4,\,2)\) - mean equals the last value (martingale), variance equals the elapsed time.

Best forecast \(\E[B_3\mid\F_1]=B_1=0.4\); forecast standard deviation \(\sqrt2\approx1.414\).

This martingale-plus-\(\sqrt{\text{time}}\)-uncertainty structure is exactly how a driftless price is projected forward.

Interactive: generate and inspect Brownian paths

Common Mistakes to Avoid

Writing \(\Var(B_t)=t^2\) or \(\sigma t\); for standard BM \(\Var(B_t)=t\), standard deviation \(\sqrt t\).
Assuming \(dB_t/dt\) exists - it does not; BM has no derivative, which is why we use stochastic, not ordinary, calculus.
Using \(\Cov(B_s,B_t)=st\); the correct covariance is \(\min(s,t)\).
Treating overlapping increments as independent; only non-overlapping increments are independent.

Quant Practitioner Tips

Remember the trio: mean 0, variance \(t\), covariance \(\min(s,t)\) - they define BM up to its Gaussian law.
Model a driftless log-price as \(\sigma B_t\): variance \(\sigma^2 t\), the source of the \(\sigma\sqrt t\) in Black-Scholes.
When a computation needs a derivative of \(B\), stop - reach for Ito’s lemma and quadratic variation instead.

Knowledge Check

Q1 Medium

For standard Brownian motion, \(\Cov(B_s,B_t)\) with \(s\le t\) equals:

\(st\)

\(\min(s,t)=s\)

\(t-s\)

\(0\)

Q2 Medium

Brownian sample paths are:

Smooth and differentiable

Continuous everywhere but differentiable nowhere

Discontinuous with jumps

Piecewise linear

Q3 Hard

Donsker’s theorem states that the rescaled random walk \(S_{\lfloor nt\rfloor}/\sqrt n\):

Diverges

Converges in distribution to Brownian motion

Converges to a straight line

Becomes a Poisson process

Practical Exercise

Let \((B_t)\) be standard Brownian motion. (a) Find the distribution of \(B_2-B_5\) is asked incorrectly - instead find the law of \(B_5-B_2\). (b) Compute \(\E[B_2 B_5]\). (c) Are \(B_2\) and \(B_5-B_2\) independent? Use this to recompute (b).

▶ Show full solution

(a) \(B_5-B_2\sim\Normal(0,5-2)=\Normal(0,3)\) by stationary Gaussian increments.

(b) \(\E[B_2 B_5]=\Cov(B_2,B_5)=\min(2,5)=2\) (both mean 0).

(c) Yes: \(B_2\) is a function of increments up to time 2, and \(B_5-B_2\) is the increment on \((2,5]\); non-overlapping increments are independent. Then \(\E[B_2 B_5]=\E[B_2(B_2+(B_5-B_2))]=\E[B_2^2]+\E[B_2]\E[B_5-B_2]=2+0=2\), matching (b).

After the reveal, answer for yourself: Why does the answer \(\min(s,t)\) equal the variance of the earlier time?

Lesson Summary

Brownian motion is the continuous-time scaling limit of the random walk, defined by four axioms: start at 0, independent and stationary Gaussian increments \(B_t-B_s\sim\Normal(0,t-s)\), and continuous paths. Its structure is fixed by mean 0, variance \(t\), and covariance \(\min(s,t)\). The paths are continuous everywhere but differentiable nowhere - the paradox that forces stochastic calculus.

Formula Sheet Additions

BM moments

\[\E[B_t]=0,\ \Var(B_t)=t,\ \Cov(B_s,B_t)=\min(s,t)\]

The defining second-order structure.

Increment law

\[B_t-B_s\sim\Normal(0,t-s),\ \text{independent of }\F_s\]

Independent stationary Gaussian increments.

Error Log Checklist

Did I use \(\Var(B_t)=t\) (not \(t^2\) or \(\sigma t\))?
Did I use \(\Cov=\min(s,t)\), and only treat non-overlapping increments as independent?
Did I resist differentiating \(B\)?

Retrieval Practice

Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.

▶ Show retrieval prompts & answers

Q: List the four defining properties of standard Brownian motion.
A: \(B_0=0\); independent increments; \(B_t-B_s\sim\Normal(0,t-s)\) (stationary Gaussian); almost-surely continuous paths.

Q: Give \(\E[B_t]\), \(\Var(B_t)\), and \(\Cov(B_s,B_t)\).
A: 0, \(t\), and \(\min(s,t)\).

Q: Why are Brownian paths not differentiable?
A: Increments scale like \(\sqrt h\), so difference quotients \((B_{t+h}-B_t)/h\sim1/\sqrt h\to\infty\); no slope exists.

Flashcards

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Brownian covariance

\(\Cov(B_s,B_t)=\min(s,t)\); mean 0, variance \(t\).

Path regularity

Continuous everywhere, differentiable nowhere, infinite variation.

Completion Checklist

I can explain the core ideas in my own words
I worked the derivations/examples by hand
I completed the interactive workbench(es)
I passed the knowledge check

Confidence / mastery rating

Personal notes

Source References

This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.

Brownian Motion and Stochastic Calculus (Karatzas & Shreve, 2nd ed., GTM 113) foundational - Ch. 2 - Ch. 2: construction and defining properties of Brownian motion.
Probability: Theory and Examples (Rick Durrett, 5th ed.) current - Ch. 7 - Ch. 7: Brownian motion, Donsker’s theorem, path properties.

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