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 |
2 Informal proof 3 Formal proof |
Then:
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
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.Informal proof
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.