Stochastic Calculus Notes 02 | Random Walks, Quadratic Variation and Brownian Motion

21 Oct 2026 • Ziyuan Zhang

Overview

Brownian motion is easiest to understand as the limit of a coin-tossing game. This post starts from the random walk, introduces the one quantity that makes stochastic calculus different from ordinary calculus — quadratic variation — and then shows that Brownian motion inherits everything: it is a martingale, \(W_t^2-t\) is a martingale, and \([W,W]_t=t\).

The punchline to keep in mind: \((dW_t)^2=dt\). Ordinary smooth curves have zero quadratic variation; Brownian paths do not, and that single fact is where Itô’s extra term will come from.

1. The random walk

Take the infinite fair coin-toss space, \(\omega=(\omega_1,\omega_2,\dots)\), with independent tosses. Define

\[Z_j(\omega)=\begin{cases}+1,&\omega_j=H\\-1,&\omega_j=T\end{cases},\qquad M_0=0,\quad M_n=\sum_{j=1}^nZ_j\ (n\ge1).\]

The \(Z_j\) are i.i.d. with \(E Z_j=0\), \(\mathrm{Var}(Z_j)=E Z_j^2=1\). \(M\) is the random walk and the \(Z_n=M_n-M_{n-1}\) are its increments.

Properties.

\(M\) is a martingale. \(\mathcal F_n\) = first \(n\) tosses; \(Z_{n+1}\) only depends on toss \(n+1\), so \(Z_{n+1}\perp\!\!\!\perp\mathcal F_n\):
\[E[M_{n+1}\mid\mathcal F_n]=E[M_n+Z_{n+1}\mid\mathcal F_n]=M_n+E[Z_{n+1}]=M_n.\]
Independent increments. For \(0\le m<n\le p<q\), \(M_n-M_m=\sum_{j=m+1}^nZ_j\) and \(M_q-M_p=\sum_{j=p+1}^qZ_j\) use disjoint tosses, so they are independent.
Increment mean and variance. \(E[M_n-M_m]=0\) and
\[\mathrm{Var}(M_n-M_m)=\sum_{i,j=m+1}^nE[Z_iZ_j]=n-m,\qquad E[Z_iZ_j]=\begin{cases}1,&i=j\\0,&i\ne j.\end{cases}\]

2. The squared random walk

\(M_n^2\) is not a martingale — it drifts up:

\[E[M_{n+1}^2\mid\mathcal F_n]=E[(M_n+Z_{n+1})^2\mid\mathcal F_n]=M_n^2+2M_nE[Z_{n+1}]+E[Z_{n+1}^2]=M_n^2+1.\]

So \(M^2\) is a sub-martingale, and subtracting the drift fixes it: \(Y_n:=M_n^2-n\) satisfies \(E[Y_{n+1}\mid\mathcal F_n]=M_n^2+1-(n+1)=Y_n\), a martingale.

3. Quadratic variation (discrete time)

For a process \(X=\{X_n\}\), the quadratic variation is the process

\[[X,X]_0=0,\qquad [X,X]_n=\sum_{j=1}^n\big(X_j-X_{j-1}\big)^2 .\]

It adds up squared increments and measures how rough the path is. (The definition does not need \(X\) to be a martingale.)

Random walk: every increment is \(\pm1\), so \([M,M]_n=n\) for every \(\omega\) — and this holds even for an unfair coin. This is special; in general \([X,X]_n\) depends on both \(n\) and \(\omega\).
\(Y_n=M_n^2-n\): using \(M_j=M_{j-1}+Z_j\) and \(Z_j^2=1\),
\[[Y,Y]_n=\sum_{j=1}^n\big(Z_j^2+2Z_jM_{j-1}-1\big)^2=\sum_{j=1}^n4M_{j-1}^2,\]
which genuinely varies with \(\omega\).

Two facts about martingales and quadratic variation

Fact 1. If \(X\) is a martingale with \(X_0=0\), then \(\mathrm{Var}(X_n)=E\big[[X,X]_n\big]\).

Proof

Since \(E X_n=E X_0=0\), \(\mathrm{Var}(X_n)=E X_n^2\). Write \(X_n=\sum_j(X_j-X_{j-1})\) and expand the square:

\[X_n^2=\sum_{i=1}^n(X_i-X_{i-1})^2+\sum_{i\ne j}(X_i-X_{i-1})(X_j-X_{j-1}).\]

For \(i<j\), condition on \(\mathcal F_{j-1}\) (Tower: condition on the second-largest time):

\[E[(X_i-X_{i-1})(X_j-X_{j-1})]=E\big[(X_i-X_{i-1})\,E[X_j-X_{j-1}\mid\mathcal F_{j-1}]\big]=E[(X_i-X_{i-1})\cdot0]=0.\]

So the cross terms vanish and \(E X_n^2=E[X,X]_n\). For the random walk this reads \(\mathrm{Var}(M_n)=n=E[M,M]_n\).

Fact 2 (quadratic variation as compensator). If \(X\) is a martingale with \(E X_n^2<\infty\), then \(\{X_n^2-[X,X]_n\}\) is a martingale.

Proof

Let \(Y_n=X_n^2-[X,X]_n\). Then

\[Y_n-Y_{n-1}=X_n^2-X_{n-1}^2-(X_n-X_{n-1})^2=2X_{n-1}(X_n-X_{n-1}),\]

and \(E[2X_{n-1}(X_n-X_{n-1})\mid\mathcal F_{n-1}]=2X_{n-1}\cdot0=0\).

Since \([X,X]_n\) only goes up, \(X_n^2\) itself is a sub-martingale; we must compensate (subtract) \([X,X]\) to get a martingale. That is why \([X,X]\) is also called the compensator. \(M_n^2-n\) above is the special case \([M,M]_n=n\).

4. From random walk to Brownian motion

Random walk and Brownian motion share independent increments with mean \(0\) and variance equal to the time elapsed. They differ in that \(M\) lives in discrete time with integer jumps, while \(W\) lives in continuous time and takes any real value.

Scaled random walk. Speed up time by \(n\) and shrink jumps to \(1/\sqrt n\):

\[W^{(n)}_t:=\frac1{\sqrt n}M_{\lfloor nt\rfloor}.\]

Independent increments: inherited from \(M\).
Right mean and variance: for \(ns,nt\) integers,
\[E[W^{(n)}_t-W^{(n)}_s]=0,\qquad \mathrm{Var}[W^{(n)}_t-W^{(n)}_s]=\frac1n\,\mathrm{Var}[M_{nt}-M_{ns}]=\frac{nt-ns}{n}=t-s.\]
Gaussian in the limit (CLT):
\[W^{(n)}_t-W^{(n)}_s=\underbrace{\sqrt{\frac{\lfloor nt\rfloor-\lfloor ns\rfloor}{n}}}_{\to\sqrt{t-s}}\cdot\underbrace{\frac1{\sqrt{\lfloor nt\rfloor-\lfloor ns\rfloor}}\sum_{j=\lfloor ns\rfloor+1}^{\lfloor nt\rfloor}Z_j}_{\to N(0,1)\text{ by the CLT}}\ \Longrightarrow\ N(0,t-s).\]

So \(W^{(n)}\to W\): Brownian motion exists, and it is literally a coin-tossing game watched from very far away. (Be careful which “\(n\)” the CLT is applied to — it is the number of tosses in the window, \(\lfloor nt\rfloor-\lfloor ns\rfloor\).)

5. Brownian motion and its properties

Recall the definition from Part 01: \(W_0=0\); continuous paths; \(W_t-W_s\sim N(0,t-s)\); independent increments. \(W\) is a BM with respect to \(\mathbb F\) if additionally it is adapted and \(W_t-W_s\perp\!\!\!\perp\mathcal F_s\).

1. Martingale. Shown in Part 01. In continuous time, super/sub-martingales and the monotonicity of \(E X_t\) work exactly as in discrete time.

2. Covariance. For \(s<t\), add and subtract \(W_s\) to create an independent increment:

\[\mathrm{Cov}(W_s,W_t)=E[W_sW_t]=E[W_s(W_t-W_s)]+E[W_s^2]=0+s=\min(s,t).\]

So \((W_s,W_t)\) is jointly normal with

\[\begin{pmatrix}W_s\\W_t\end{pmatrix}\sim N\left(\begin{pmatrix}0\\0\end{pmatrix},\begin{pmatrix}s&s\\s&t\end{pmatrix}\right).\]

3. Conditional law and the Markov property. Using \(E[e^{\alpha Y}]=e^{\alpha\mu+\frac12\alpha^2\sigma^2}\) for \(Y\sim N(\mu,\sigma^2)\):

\[E[e^{\alpha W_t}\mid\mathcal F_s]=e^{\alpha W_s}\,E[e^{\alpha(W_t-W_s)}]=e^{\alpha W_s+\frac12\alpha^2(t-s)}\ \Longrightarrow\ W_t\mid\mathcal F_s\sim N(W_s,\ t-s).\]

Given everything up to time \(s\), the law of \(W_t\) depends only on \(W_s\) — the Markov property. The transition density is

\[p(\tau,y,x)=\frac1{\sqrt{2\pi\tau}}\,e^{-\frac{(x-y)^2}{2\tau}},\qquad E[h(W_{s+\tau})\mid W_s=y]=\int_{\mathbb R}h(x)\,p(\tau,y,x)\,dx,\]

read as “go from \(y\) to near \(x\) in time \(\tau\)”.

4. \(W_t^2-t\) is a martingale. With \(W_t^2=(W_t-W_s)^2+2(W_t-W_s)W_s+W_s^2\):

\[E[W_t^2-t\mid\mathcal F_s]=(t-s)+0+W_s^2-t=W_s^2-s.\]

This is the continuous-time version of \(M_n^2-n\).

6. Quadratic variation in continuous time

Fix \(t\) and a partition \(\Pi=\{0=t_0<t_1<\dots<t_n=t\}\) with mesh \(\Vert \Pi\Vert =\max_j(t_j-t_{j-1})\). Define

\[[X,X]^\Pi_t=\sum_{j=1}^n\big(X_{t_j}-X_{t_{j-1}}\big)^2,\qquad [X,X]_t=\lim_{\Vert \Pi\Vert \to0}[X,X]^\Pi_t .\]

Over a window, \([X,X]_{[s,t]}\) is defined the same way and quadratic variation accumulates: \([X,X]_t=[X,X]_s+[X,X]_{[s,t]}\).

Theorem. \([W,W]_t=t\) with probability one, and \([W,W]_{[s,t]}=t-s\).

Intuition. For the scaled walk, \([W^{(n)},W^{(n)}]_t=\frac1n\lfloor nt\rfloor\in[t-\frac1n,t]\), and \(W^{(n)}\to W\).

Proof sketch

Take the uniform partition \(t_j=jt/n\). Then

\[[W,W]^\Pi_t=\frac tn\sum_{j=1}^n\left(\frac{W_{t_j}-W_{t_{j-1}}}{\sqrt{t_j-t_{j-1}}}\right)^2=t\cdot\frac1n\sum_{j=1}^nZ_j^2,\]

with \(Z_j\) i.i.d. \(N(0,1)\). By the strong law of large numbers, \(\frac1n\sum Z_j^2\to E[Z^2]=1\), so \([W,W]^\Pi_t\to t\). For a general partition one shows \(E\big[[W,W]^\Pi_t\big]=t\) and \(\mathrm{Var}\big[[W,W]^\Pi_t\big]\le K\Vert \Pi\Vert\) with \(K\) depending only on \(t\), which forces convergence to \(t\).

Fact 2 carries over: if \(X\) is a continuous martingale with \(E X_t^2<\infty\), then \(X_t^2-[X,X]_t\) is a martingale (proof needs Itô’s formula, Part 04).

7. Why Brownian paths must be rough

The first-order variation of a path is \(FV(X)_t=\lim_{\Vert \Pi\Vert \to0}\sum_j\vert X_{t_j}-X_{t_{j-1}}\vert\). For a smooth path, the mean value theorem gives

\[FV(X)_t=\lim\sum_j(t_j-t_{j-1})\vert X'(\xi_j)\vert=\int_0^t\vert X'_s\vert\,ds<\infty.\]

Finite first variation forces zero quadratic variation (for continuous paths):

\[\sum_j(X_{t_j}-X_{t_{j-1}})^2\le\underbrace{\max_j\vert X_{t_j}-X_{t_{j-1}}\vert}_{\to0\ \text{(continuity)}}\cdot\underbrace{\sum_j\vert X_{t_j}-X_{t_{j-1}}\vert}_{\to FV(X)_t<\infty}\ \to0.\]

Since \([W,W]_t=t>0\), Brownian paths cannot have finite first variation, and in fact they are nowhere differentiable with probability one.

This roughness is a feature, not a bug: a smooth price path would be locally predictable, and predictability means arbitrage.

The heuristic everyone uses. Differentiating \([W,W]_t=t\) gives \(d[W,W]_t=dt\), and people write

\[(dW_t)^2=dt,\]

thinking of \([W,W]_{[t,t+dt]}\approx\sum(W_{t_j}-W_{t_{j-1}})^2\). Taking a square root to say “\(dW_t=\sqrt{dt}\)” is meaningless, but the multiplication rule \((dW_t)^2=dt\) is exactly right and drives all of Part 04.

Series contents

01 · Probability Spaces, Filtrations, Conditional Expectation and Martingales
02 · Random Walks, Quadratic Variation and Brownian Motion (this post)
03 · Stochastic Integration
04 · Itô’s Formula, Itô Processes and Geometric Brownian Motion
05 · Black–Scholes: Replication, the PDE and the Greeks
06 · Risk-Neutral Pricing, Change of Measure and Girsanov
07 · SDEs, Feynman–Kac and Stopping Times

← Previous: Probability Spaces, Filtrations, Conditional Expectation and Martingales　·　Next: Stochastic Integration →

This series collects my notes from QST MF 790 Stochastic Calculus at Boston University’s Questrom School of Business, in 7 parts. The content follows the order of the course; wording, intuition and a few corrections are my own. Throughout, \(N(\mu,\sigma^2)\) uses the variance as its second argument. Corrections are welcome. Chinese version of this post: 中文版.