Let <math>x(t)</math> be an Ito (or Generalized Wiener) process[?]. That is letand let f be some function with a second derivative that is continuous. Then:
 <math> x(t) = a(x,t)dt + b(x,t)dW_t</math>
 <math> f(x(t)) </math> is also an Ito process.
 <math> df(x(t),t) = ( a(x,t)\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t} + \frac{b(x,t)*b(x,t)* \frac{\partial^2f}{\partial x^2}}{2})dt + b(x,t)\frac{\partial f}{dx}dW_t</math>
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 <math>dt^2</math> and <math>dt*dW</math> terms disappear but the <math>dW^2</math> tends to dt. Substituting this dt in, and reordering the terms so that the dt and dW terms are collected we obtain
as required.
A strongwilled individual is required here!
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