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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a lemma is a proven statement, typically named as such to distinguish it as a truth used as a stepping stone to a larger result rather than an important statement in and of itself. ... Stochastic calculus is a branch of mathematics that operates on stochastic processes. ... In mathematics differential has various meanings. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In the mathematics of probability, a stochastic process can be thought of as a random function. ... In calculus, the chain rule is a formula for the derivative of the composition of two functions. ... For other uses of the term calculus see calculus (disambiguation) Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ... Financial mathematics is the branch of applied mathematics concerned with the financial markets. ...

Statement of the lemma

Let x(t) be an Itô (or generalized Wiener) process. That is let Itô calculus, named after Kiyoshi Itô, treats mathematical operations on stochastic processes. ...

and let f be some function with a second derivative that is continuous. In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

Then:

f(x(t),t) is also an Itô process.

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. In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true. ...

Expanding f(x, t) in a Taylor series in x and t we have As the degree of the taylor series rises, it approaches the correct function. ...

and substituting in for dx from above we have

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

, as

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

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. Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... Probability theory is the mathematical study of probability. ...

See also

• Wiener process
• Itô calculus

In mathematics, the Wiener process, so named in honor of Norbert Wiener, is a continuous-time Gaussian stochastic process with independent increments used in modelling Brownian motion and some random phenomena observed in finance. ... Itô calculus, named after Kiyoshi Itô, treats mathematical operations on stochastic processes. ...

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), optiontutor