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## Encyclopedia > Ito's Lemma

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# Ito's Lemma

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.
Let $x(t)$ be an Ito (or Generalized Wiener) process[?]. That is let
$x(t) = a(x,t)dt + b(x,t)dW_t$
and let f be some function with a second derivative that is continuous. Then:
$f(x(t))$ is also an Ito process.
$df(x(t),t) = ( a(x,t)\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t} + \frac{b(x,t)*b(x,t)* \frac{\partial^2f}{\partial x^2}}{2})dt + b(x,t)\frac{\partial f}{dx}dW_t$

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

$df = \frac{\partial f}{\partial x}{dx} + \frac{\partial f}{\partial t}dt + \frac{1}{2}\frac{\partial^2 f}{\partial x^2}dx^2+ ...$

and substituting in for dx from above we have

$df = \frac{\partial f}{\partial x}(a.dt + b.dW_t) + \frac{\partial f}{\partial t}dt + \frac{1}{2}\frac{\partial^2 f}{\partial x^2}(a^2(dt)^2 + 2.a.b.dt.dW + b^2(dW^2))+ ...$

In the limit as dt tends to 0 the $dt^2$ and $dt*dW$ terms disappear but the $dW^2$ tends to dt. Substituting this dt in, and reordering the terms so that the dt and dW terms are collected we obtain

$df = ( a\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t} + \frac{b*b*\frac{\partial^2f}{\partial x^2}}{2})dt + b\frac{\partial f}{dx}dW_t$

as required.

A strong-willed individual is required here!

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