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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 <math>x(t)</math> be an Ito (or Generalized Wiener) process[?]. That is let
 <math> x(t) = a(x,t)dt + b(x,t)dW_t</math>
and let f be some function with a second derivative that is continuous.
Then:
 <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>
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
 <math> df = \frac{\partial f}{\partial x}{dx} + \frac{\partial f}{\partial t}dt + \frac{1}{2}\frac{\partial^2 f}{\partial x^2}dx^2+ ...</math>
and substituting in for dx from above we have
 <math> df = \frac{\partial f}{\partial x}(a.dt + b.dW_t) + \frac{\partial f}{\partial t}dt + \frac{1}{2}\frac{\partial^2 f}{\partial x^2}(a^2(dt)^2 + 2.a.b.dt.dW + b^2(dW^2))+ ...</math>
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
 <math> df = ( a\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t} + \frac{b*b*\frac{\partial^2f}{\partial x^2}}{2})dt + b\frac{\partial f}{dx}dW_t</math>
as required.
Formal proof
A strongwilled individual is required here!
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