Itō's lemma

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In mathematics, Itō's lemma is used in stochastic calculus to find the differential of a function of a particular type of stochastic process. It is the stochastic calculus counterpart of the chain rule in ordinary calculus and is best memorized using the Taylor series expansion and retaining the second order term related to the stochastic component change. The lemma is widely employed in mathematical finance.

1 Statement of the lemma
2 Informal proof
3 See also
4 References
5 External links

[edit] Statement of the lemma

Let x(t) be an Itō (or generalized Wiener) process. That is let

$dx(t) = a(x,t)\,dt + b(x,t)\,dW_t$

where W_t is a Wiener process, and let f(x, t) be a function with continuous second derivatives.

Then $f (x (t), t)$ is also an Itō process, and

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

This is not Ito's Lemma, and is in fact just a specialization of the Lemma.

[edit] Informal proof

A formal proof of the lemma requires us to take the limit of a sequence of random variables, which is not done here. Expanding f(x, t) in a Taylor series in x and t we have

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

and substituting a dt + b dW for dx gives

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

In the limit as dt tends to 0, the dt² and dt dW terms disappear but the dW² term tends to dt. The latter can be shown if we prove that

$dW^2 \rightarrow E(dW^2),$ since $E(dW^2) = dt. \,$

The proof of this statistical property is however beyond the scope of this article.

Deleting the dt² and dt dW terms, substituting dt for dW², and collecting the dt and dW terms, we obtain

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

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.

[edit] See also

[edit] References

Kiyoshi Itō (1951). On stochastic differential equations. Memoirs, American Mathematical Society 4, 1–51.
Hagen Kleinert (2004). Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edition, World Scientific (Singapore); Paperback ISBN 981-238-107-4. Also available online: PDF-files. This textbook also derives generalizations of Itō's lemma for non-Wiener (non-Gaussian) processes.
Bernt Øksendal (2000). Stochastic Differential Equations. An Introduction with Applications, 5th edition, corrected 2nd printing. Springer. ISBN 3-540-63720-6. Sections 4.1 and 4.2.