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In mathematics, Itô's lemma is 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.
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Statement of the lemma
Let x(t) be an Itô (or generalized Wiener) process. That is let
- <math> dx(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),t) <math> is also an Itô process.
- <math> df(x(t),t) = \left( 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} \right) 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) in 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+ \cdots<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))+ \cdots<math>
In the limit as dt tends to 0 the dt2 and dt dW terms disappear but the dW2 tends to dt. The latter can be shown if we prove that
- <math> dW^2 = E\left(dW^2\right)<math>, as <math>E\left(dW^2\right) = dt.<math>
The proof of this statistical property is however beyond the scope of this article.
Substituting this dt in, and reordering the terms so that the dt and dW terms are collected we obtain
- <math> df = \left( a\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t} + \frac{b*b*\frac{\partial^2f}{\partial x^2}}{2}\right) dt + b\frac{\partial f}{dx}\,dW_t<math>
as required.
The formal proof, which is not included in this article, requires defining the stochastic integral, which is an advanced concept in between functional analysis and probability theory.
See also
External links
- Derivation (http://www2.sjsu.edu/faculty/watkins/ito.htm), Prof. Thayer Watkins
- Discussion (http://www.quantnotes.com/fundamentals/backgroundmaths/ito.htm), quantnotes.com
- informal proof (http://secure.webstation.net/~ftsweb/texts/optiontutor/chap6.8.htm), optiontutorde:Lemma von Itō