generalized Ito formula
The generalized Itô formula, or generalized Itô’s lemma, is an extension of Itô’s lemma (http://planetmath.org/ItosLemma2) that applies also to discontinuous processes. For a cadlag process , we write for its jump at time .
This is indeed a well defined finite variation process, as the sum of the absolute values
The reason for using differential notation and writing the formula in terms of the continuous part of the quadratic covariation should be clear when it is considered that writing out the expression in full gives the following rather messy formula.
This formula may be understood as Itô’s formula (http://planetmath.org/ItosLemma2) for continuous processes together with an additional term to ensure that the jumps of the right hand side are equal to . The need for this adjustment term comes from the fact that Itô’s formula for continuous processes is essentially a Taylor expansion to second order, which only applies when the increments vanish in the limit of small . This needs adjusting whenever the process jumps.
Suppose that is a semimartingale taking values in an open subset of and is twice continuously differentiable. Then is a semimartingale.
|Title||generalized Ito formula|
|Date of creation||2013-03-22 18:41:47|
|Last modified on||2013-03-22 18:41:47|
|Last modified by||gel (22282)|
|Synonym||generalized Itô formula|
|Synonym||generalized Itö formula|
|Synonym||generalized Ito’s lemma|
|Synonym||generalized Itô’s lemma|
|Synonym||generalized Itö’s lemma|