Chapter 06

Hard

Itô's Lemma

00 · Symbol Glossary

$f(W_t,t)$a smooth function of Brownian motion and time

Assumed $C^2$ in the spatial argument and $C^1$ in time — enough derivatives to Taylor-expand, which is all Itô's Lemma is: a Taylor expansion kept honest about quadratic variation.

$[X]_t$quadratic variation of $X$

From Chapter 03: for Brownian motion, $[W]_t=t$ ; informally $(dW_t)^2=dt$ . This replacement is the entire mechanical content of Itô's Lemma.

$dX_t = \mu_t\,dt + \sigma_t\,dW_t$a general Itô process

Shorthand for $X_t = X_0+\int_0^t \mu_s\,ds+\int_0^t \sigma_s\,dW_s$ , the object Chapter 05 built. $\mu_t$ is the drift, $\sigma_t$ the diffusion coefficient.

$f_x,\,f_{xx},\,f_t$partial derivatives of $f$

Standard subscript notation: $f_x=\partial f/\partial x$ , $f_{xx}=\partial^2 f/\partial x^2$ , $f_t=\partial f/\partial t$ . Itô's Lemma is a formula for $df$ in terms of exactly these.

01 · Taylor's Theorem, Kept Honest

Ordinary calculus differentiates $f(x_t)$ by a first-order Taylor expansion: $df=f'(x_t)\,dx_t$ , with second-order terms discarded because $(dx_t)^2$ is negligible compared to $dx_t$ when $x_t$ is smooth. Chapter 03 showed that is false for Brownian motion — $(\Delta W)^2$ is the same order as $\Delta t$ , not smaller. Itô's Lemma is the second-order Taylor expansion with that fact kept instead of discarded.

Start from a second-order expansion of $f(W_{t+\Delta t})-f(W_t)$ over a small step:

\Delta f \approx f'(W_t)\,\Delta W + \tfrac12 f''(W_t)\,(\Delta W)^2 + O((\Delta W)^3)

In ordinary calculus, $(\Delta W)^2$ would be $O(\Delta t^2)$ and vanish faster than $\Delta t$ as the partition is refined. For Brownian motion, $\mathbb{E}[(\Delta W)^2]=\Delta t$ , and Chapter 03's quadratic-variation result says $\sum(\Delta W)^2\to t$ — the sum of squared increments does not vanish, it converges to a finite, nonzero limit. The correct replacement, justified by that convergence, is $(dW_t)^2=dt$ inside any expression that is eventually integrated.

The one rule to memorize

$(dW_t)^2=dt$ , $\,dW_t\,dt=0$ , $\,(dt)^2=0$ . Every version of Itô's Lemma below is Taylor's theorem plus this substitution rule, nothing more.

02 · Itô's Lemma for $f(W_t,t)$

Definition — Itô's Lemma, Pure Brownian Case

If $f(x,t)$ is $C^2$ in $x$ and $C^1$ in $t$ , then

df(W_t,t) = f_t(W_t,t)\,dt + f_x(W_t,t)\,dW_t + \tfrac12 f_{xx}(W_t,t)\,dt

equivalently, in integral form,

f(W_T,T) = f(W_0,0) + \int_0^T \left(f_t+\tfrac12 f_{xx}\right)dt + \int_0^T f_x\,dW_t

Plain language: the ordinary chain-rule term $f_x\,dW_t$ is still there, but there is an extra drift term $\tfrac12 f_{xx}\,dt$ that has no analogue in deterministic calculus. It appears because the second-order Taylor term $\tfrac12 f_{xx}(dW_t)^2$ does not vanish — it becomes $\tfrac12 f_{xx}\,dt$ , a bona fide first-order contribution to the drift.

Example — $d(W_t^2)$

Let $f(x)=x^2$ , so $f_x=2x$ , $f_{xx}=2$ . Itô's Lemma gives

d(W_t^2) = 2W_t\,dW_t + \tfrac12(2)\,dt = 2W_t\,dW_t + dt

Integrating: $W_T^2 = \int_0^T 2W_t\,dW_t + T$ , i.e. $\int_0^T W_t\,dW_t=\tfrac12 W_T^2-\tfrac12 T$ — exactly the identity flagged in Chapter 05's FailBlock, now derived rather than asserted.

03 · General Diffusions

The pure-Brownian case extends immediately once $W_t$ is replaced by a general Itô process $X_t$ .

Definition — Itô's Lemma, General Case

Let $dX_t = \mu_t\,dt+\sigma_t\,dW_t$ and let $f(x,t)$ be $C^2$ in $x$ , $C^1$ in $t$ . Then

df(X_t,t) = \left(f_t + \mu_t f_x + \tfrac12 \sigma_t^2 f_{xx}\right)dt + \sigma_t f_x\,dW_t

Plain language: $f_x$ carries through both the drift $\mu_t$ and the noise $\sigma_t\,dW_t$ exactly as the ordinary chain rule would predict, but there is again a correction $\tfrac12 \sigma_t^2 f_{xx}\,dt$ coming from $(dX_t)^2=\sigma_t^2\,dt$ — the increment squared inherits its quadratic variation entirely from the diffusion coefficient, not the drift, because $dt$ and $dW_t\,dt$ terms vanish under squaring while $(\sigma_t\,dW_t)^2=\sigma_t^2\,dt$ survives. This is the formula that will turn the SDE for a stock price (Chapter 08) into a solvable equation for $\log S_t$ .

Step-by-step — Applying Itô's Lemma

Identify $X_t$ 's drift and diffusion: write

dX_t=\mu_t\,dt+\sigma_t\,dW_t

explicitly; misreading either coefficient propagates into every term below.

Compute $f_t$ , $f_x$ , $f_{xx}$ : ordinary partial derivatives of the transform

f

, evaluated at

(X_t,t)

Assemble the drift:

f_t+\mu_t f_x+\tfrac12\sigma_t^2 f_{xx}

— three separate contributions, and it is easy to forget the third.

Assemble the diffusion term:

\sigma_t f_x\,dW_t

— this one matches ordinary chain-rule intuition exactly.

04 · Worked Examples

Example — $d(e^{W_t})$

$f(x)=e^x$ , so $f_x=f_{xx}=e^x$ . With $\mu_t=0,\sigma_t=1$ (pure $W_t$ ),

d(e^{W_t}) = \tfrac12 e^{W_t}\,dt + e^{W_t}\,dW_t

Notice the drift is strictly positive: $e^{W_t}$ has an upward bias even though $W_t$ itself has zero drift. This is not a modeling choice — it falls straight out of $f_{xx}>0$ (convexity of the exponential) and is the mechanism behind the lognormal mean correction that Chapter 08 needs for geometric Brownian motion.

Example — $d(W_t^3)$ as a check of the pattern

$f(x)=x^3$ , $f_x=3x^2$ , $f_{xx}=6x$ . Then $d(W_t^3)=3W_t^2\,dW_t+3W_t\,dt$ . Unlike the two previous examples, the correction term here is itself random ( $3W_t\,dt$ , not a constant), because $f_{xx}=6x$ depends on $x$ . Constant $f_{xx}$ is the special case that produces a deterministic correction; it is not the general rule.

❌ Applying the chain rule without the second-order term

A common shortcut: " $f(W_t)$ changes by $f'(W_t)\,dW_t$ , same as ordinary calculus" — and then $d(e^{W_t})=e^{W_t}\,dW_t$ is reported with no drift.

Why it breaks: it drops the $\tfrac12 f_{xx}\,dt$ term entirely, treating $(dW_t)^2$ as negligible the way ordinary calculus treats $(dx)^2$ . Section 01 is precisely the argument for why that is wrong for Brownian motion.

Consequence: the resulting process has the wrong mean. $\mathbb{E}[e^{W_t}]$ under the correct formula grows like $e^{t/2}$ (matching the lognormal moment-generating function directly), while the shortcut formula implies a driftless martingale with constant mean $\mathbb{E}[e^{W_0}]=1$ — flatly wrong for $t>0$ .

05 · Exercises

EXERCISE 6.1

Apply the general formula with $\mu_t,\sigma_t$ constant and $f(x,t)=g(x)$ time-independent.

$df(X_t)=\left(\mu g'(X_t)+\tfrac12\sigma^2 g''(X_t)\right)dt+\sigma g'(X_t)\,dW_t$ . Setting $g(x)=x^2$ : $d(X_t^2)=(2\mu X_t+\sigma^2)\,dt+2\sigma X_t\,dW_t$ . The $\sigma^2\,dt$ term is the quadratic-variation correction; it survives even though $g''=2$ is constant, because $\sigma_t^2\neq 0$ .

For $dX_t=\mu\,dt+\sigma\,dW_t$ (constant $\mu,\sigma$ ), compute $d(X_t^2)$ using the general Itô's Lemma.

EXERCISE 6.2

$f(x)=\log x$ has $f_x=1/x$ , $f_{xx}=-1/x^2$ ; apply with $X_t=S_t$ following $dS_t=\mu S_t\,dt+\sigma S_t\,dW_t$ .

$d(\log S_t)=\left(\mu-\tfrac12\sigma^2\right)dt+\sigma\,dW_t$ . The correction $-\tfrac12\sigma^2\,dt$ is negative here because $f_{xx}=-1/S_t^2<0$ (log is concave) — the opposite sign from the $e^{W_t}$ example in Section 04, where convexity gave a positive correction. This identity is exactly the tool Chapter 08 uses to solve geometric Brownian motion.

Let $S_t$ follow $dS_t=\mu S_t\,dt+\sigma S_t\,dW_t$ . Compute $d(\log S_t)$ and explain the sign of the correction term.

EXERCISE 6.3

Compare $d(e^{W_t})$ (Section 04) to $d(e^{\sigma W_t-\sigma^2 t/2})$ ; the second is built to cancel its own correction term.

Let $Y_t=e^{\sigma W_t-\sigma^2 t/2}$ . With $f(x,t)=e^{\sigma x-\sigma^2 t/2}$ : $f_t=-\tfrac{\sigma^2}{2}f$ , $f_x=\sigma f$ , $f_{xx}=\sigma^2 f$ . So $dY_t=\left(-\tfrac{\sigma^2}{2}+\tfrac12\sigma^2\right)f\,dt+\sigma f\,dW_t=\sigma Y_t\,dW_t$ — the drift cancels exactly, leaving a pure martingale. This is the stochastic exponential, and its driftless property is the seed of Girsanov's Theorem (Chapter 10).

Show that $Y_t=e^{\sigma W_t-\sigma^2 t/2}$ has zero drift under Itô's Lemma. Why is this remarkable given that $e^{W_t}$ alone (Section 04) has strictly positive drift?

06 · Chapter Summary

Concept	Meaning
Core substitution	$(dW_t)^2=dt$ , $dW_t\,dt=0$ , $(dt)^2=0$
Pure Brownian case	$df=f_t\,dt+f_x\,dW_t+\tfrac12 f_{xx}\,dt$
General diffusion	$df=(f_t+\mu f_x+\tfrac12\sigma^2 f_{xx})\,dt+\sigma f_x\,dW_t$
Correction source	second-order Taylor term, kept instead of discarded
$d(\log S_t)$	drift shifts by $-\tfrac12\sigma^2$ relative to naive log
Common error	dropping $\tfrac12 f_{xx}\,dt$ , producing the wrong mean

Next: Chapter 07 — Stochastic Differential Equations, where Itô's Lemma becomes the tool for turning an equation for $dX_t$ into a closed-form solution for $X_t$ .