Ito's lemma

In mathematics, Ito's lemma is a lemma used in stochastic calculus to find the differential of a function of a particular type of stochastic process. It is therefore to stochastic calculus what the chain rule is to ordinary calculus. The lemma is widely employed in mathematical finance.

Table of contents

1 Statement of the lemma 2 Informal proof 3 Formal proof

Statement of the lemma

Let be an Ito (or Generalized Wiener) process. That is let

and let f be some function with a second derivative that is continuous.

Then:

is also an Ito process.

Informal proof

A formal proof of the lemma requires us to take the limit of a sequence of random variables, which is not handled carefully here.

Expanding f(x,t) is a Taylor series in x and t we have

and substituting in for dx from above we have

In the limit as dt tends to 0 the and terms disappear but the tends to dt. Substituting this dt in, and reordering the terms so that the dt and dW terms are collected we obtain

as required.

Formal proof

A strong-willed individual is required here!