Itô's Formula

Classical calculus has the chain rule: if $f$ is differentiable and $g(t)$ is differentiable, then $\frac{d}{dt}f(g(t)) = f'(g(t)) g'(t)$ . In integral form: $f(g(T)) = f(g(0)) + \int_0^T f'(g(t)) g'(t) \, dt$ .

When $g(t) = W_t$ (Brownian motion), there is no $g'(t)$ — BM paths are nowhere differentiable. Yet a chain rule still exists, with one extra term:

f(W_T) = f(W_0) + \int_0^T f'(W_t) \, dW_t + \frac{1}{2}\int_0^T f''(W_t) \, dt.

This is Itô's formula — the fundamental theorem of stochastic calculus. The extra

\frac{1}{2}f''(W_t) dt

term, the Itô correction, comes directly from the quadratic variation

[W, W]_t = t

: BM increments of size

\sqrt{dt}

contribute to the second-order Taylor term at order

dt

, unlike classical functions where

(dx)^2 = O(dt^2)

vanishes.

In finance, Itô's formula applied to $V(t, S_t)$ (an option's price as a function of time and stock price) yields the Black-Scholes PDE — the correction term generates the $\frac{1}{2}\sigma^2 S^2 \partial_{SS} V$ term.

Learning objectives. By the end of this note you should be able to:

State Itô's formula (Theorem 3.3.1) in both integral and differential form.
Verify the formula on $f(x) = x^2$ and $f(x) = e^{\sigma x}$ and connect to known martingales.
Apply the formula to compute $d(\ln S_t)$ for GBM ( $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$ ), obtaining the log-price SDE.
Explain the origin of the Itô correction term from second-order Taylor expansion plus $[W,W]_t = t$ .
Compute $\int_0^T W_t \, dW_t$ , $\int_0^T W_t^2 \, dW_t$ , and related integrals using Itô's formula.

In this lesson

The Theorem
Key Examples
General Form for Itô Processes
The Informal Rules
Computing Stochastic Integrals via Itô's Formula

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