Itô's Lemma

Brownian motion in one paragraph

A standard Brownian motion $W_t$ starts at $0$ , has independent increments, continuous paths, and $W_t - W_s \sim N(0, t - s)$ for $s < t$ .

Its increments over a short interval $\Delta t$ have size about $\sqrt{\Delta t}$ , which is much larger than $\Delta t$ . As a result, the sum of squared increments over $[0, t]$ does not vanish as the grid is refined:

\sum_i (W_{t_{i+1}} - W_{t_i})^2 \;\longrightarrow\; t .

This is the quadratic variation, often summarised by the heuristic multiplication table

(dW_t)^2 = dt, \qquad dW_t\,dt = 0, \qquad (dt)^2 = 0 .

Let $X_t$ satisfy $dX_t = \mu_t\,dt + \sigma_t\,dW_t$ and let $f(t, x)$ be twice continuously differentiable. Then

df(t, X_t) = \frac{\partial f}{\partial t}\,dt + \frac{\partial f}{\partial x}\,dX_t + \frac12\,\frac{\partial^2 f}{\partial x^2}\,(dX_t)^2 ,

and with $(dX_t)^2 = \sigma_t^2\,dt$ ,

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

It is a second-order Taylor expansion in which the second-order term survives because $(dW)^2$ is of order $dt$ .

Worked example

solving geometric Brownian motion

A stock follows $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$ . Find $S_t$ explicitly.

Show solution

Apply Itô to $f(S) = \ln S$ , with $f' = 1/S$ and $f'' = -1/S^2$ :

d\ln S_t = \frac{1}{S_t}\,dS_t - \frac{1}{2S_t^2}\,\sigma^2 S_t^2\,dt = \left(\mu - \tfrac12\sigma^2\right)dt + \sigma\,dW_t .

The right-hand side has constant coefficients, so integrate directly:

S_t = S_0 \exp\!\Big(\big(\mu - \tfrac12\sigma^2\big)t + \sigma W_t\Big).

The $-\tfrac12\sigma^2$ is the convexity correction: the median of $S_t$ grows at $\mu - \tfrac12\sigma^2$ while its mean grows at $\mu$ .

Knowledge check

Using Itô's lemma, what is the $dt$ coefficient of $d(W_t^3)$ ?

Knowledge check

For which value of $c$ is $M_t = e^{W_t - c\,t}$ a martingale?

Practice problems

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Answer the 2 knowledge checks to complete this lesson (0 done).