Stochastic calculus

Itô’s lemma

Ordinary calculus plus one extra term, because $(dW)^2 = dt$ rather than zero.

df = \left(f_t + \mu f_x + \tfrac12\sigma^2 f_{xx}\right)dt + \sigma f_x\,dW

Where

$\tfrac12\sigma^2 f_{xx}$: The Itô term. Dropping it is the commonest error in the subject.
$(dW)^2 = dt$: Quadratic variation of Brownian motion is deterministic.

Assumptions

$dX = \mu\,dt + \sigma\,dW$ , with $f$ twice differentiable in $x$ and once in $t$ .

Sanity check. Apply it and check whether the $dt$ term vanishes — that is the whole martingale test.

Where this is taught

Itô’s lemma and the computations you will be asked for · SC · The Itô integral and Itô’s lemma

Itô’s lemma · QuantMax