stochastic integration by parts
The stochastic integral satisfies a version of the classical integration by parts formula, which is just the integral version of the product rule. The only difference here is the existence of a quadratic covariation term.
Let be semimartingales. Then,
Alternatively, in differential notation, this reads
The existence of the quadratic covariation term in the integration by parts formula, and also in Itô’s lemma, is an important difference between standard calculus and stochastic calculus. To see the need for this term, consider the following. Choosing any , write the increment of a process over a time step of size as . The increment of a product of processes satisfies the following simple identity,
As we let tend to zero, for differentiable processes the final term of (2) is of order (http://planetmath.org/LandauNotation) , so can be neglected in the limit. However, when and are semimartingales, such as Brownian motion, the final term will be of order , and needs to be retained even in the limit.
The proof of equation (1) is given by the proof of the existence of the quadratic variation of semimartingales (http://planetmath.org/QuadraticVariationOfASemimartingale) and, in particular, is just a rearrangement of the formula given for the quadratic covariation of semimartingales. Whenever either of or is a continuous finite variation process, the quadratic covariation term is zero, so (1) becomes the standard integration by parts formula. More generally, for noncontinuous processes we have the following.
Let be a semimartingale and be an adapted finite variation process. Then,
As is a finite variation process, the first integral on the right hand side of (3) makes sense as a Lebesgue-Stieltjes integral. Equation (3) follows from the integration by parts formula by first substituting the following formula for the covariation whenever has finite variation into (1)
and then using the following identity
|Title||stochastic integration by parts|
|Date of creation||2013-03-22 18:41:35|
|Last modified on||2013-03-22 18:41:35|
|Last modified by||gel (22282)|