Itô’s lemma, also known as Itô’s formula, is an extension of the chain rule (http://planetmath.org/ChainRuleSeveralVariables) to the stochastic integral, and is often regarded as one of the most important results of stochastic calculus. The case described here applies to arbitrary continuous semimartingales. For the application to Itô processes see Itô’s formula (http://planetmath.org/ItosFormula) or see the generalized Itô formula (http://planetmath.org/GeneralizedItoFormula) for noncontinuous processes.
In particular, for a continuous real-valued semimartingale , (1) becomes
which is a form of the “change of variables formula” for stochastic calculus. A major distinction between standard and stochastic calculus is that here we need to include the quadratic variation and covariation terms and .
Taking the limit as goes to zero, all of the terms on the right hand side of (2), other than the first, go to zero with order (http://planetmath.org/LandauNotation) and, therefore, can be neglected in the limit. This results in the standard chain rule. However, when for a semimartingale then the second order terms in (2) only go to zero at rate and, therefore, must be retained even in the limit as . This is a consequence of semimartingales, such as Brownian motion, being nowhere differentiable. In fact, if is a finite variation process, then it can be shown that the quadratic covariation terms are zero, and the standard chain rule results.
A consequence of Itô’s lemma is that if is a continuous semimartingale and is twice continuously differentiable, then will be a semimartingale. However, the generalized Itô formula shows that it is not necessary to restrict this statement to continuous processes.
|Date of creation||2013-03-22 18:41:44|
|Last modified on||2013-03-22 18:41:44|
|Last modified by||gel (22282)|