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# 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.

## 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!