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.

Theorem.

Let $X,Y$ be semimartingales. Then,

 $X_{t}Y_{t}=X_{0}Y_{0}+\int_{0}^{t}X_{s-}\,dY_{s}+\int_{0}^{t}Y_{s-}\,dX_{s}+[X% ,Y]_{t}.$ (1)

Alternatively, in differential notation, this reads

 $d(X_{t}Y_{t})=X_{t-}dY_{t}+Y_{t-}dX_{t}+d[X,Y]_{t}.$

The existence of the quadratic covariation term $[X,Y]$ in the integration by parts formula, and also in ItΓ΄βs lemma, is an important difference between standard calculus and stochastic calculus. To see the need for this term, consider the following. Choosing any $h>0$, write the increment of a process over a time step of size $h$ as $\delta X_{t}\equiv X_{t+h}-X_{t}$. The increment of a product of processes satisfies the following simple identity,

 $\delta(XY)_{t}=X_{t}\delta Y_{t}+Y_{t}\delta X_{t}+\delta X_{t}\delta Y_{t}.$ (2)

As we let $h$ tend to zero, for differentiable processes the final term of (2) is of order (http://planetmath.org/LandauNotation) $O(h^{2})$, so can be neglected in the limit. However, when $X$ and $Y$ are semimartingales, such as Brownian motion, the final term will be of order $h$, 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 $X$ or $Y$ is a continuous finite variation process, the quadratic covariation term $[X,Y]$ is zero, so (1) becomes the standard integration by parts formula. More generally, for noncontinuous processes we have the following.

Corollary.

Let $X$ be a semimartingale and $Y$ be an adapted finite variation process. Then,

 $X_{t}Y_{t}=X_{0}Y_{0}+\int_{0}^{t}X_{s}\,dY_{s}+\int_{0}^{t}Y_{s-}\,dX_{s}.$ (3)

As $Y$ 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 $Y$ has finite variation into (1)

 $[X,Y]_{t}=\sum_{s\leq t}\Delta X_{s}\Delta Y_{s}$

and then using the following identity

 $\begin{split}\displaystyle\int_{0}^{t}X_{s}\,dY_{s}-\int_{0}^{t}X_{s-}\,dY_{s}% &\displaystyle=\int_{0}^{t}\Delta X_{s}\,dY_{s}=\int_{0}^{t}\sum_{u}\Delta X_{% u}1_{\{u=s\}}\,dY_{s}\\ &\displaystyle=\sum_{u\leq t}\Delta X_{u}\Delta Y_{u}.\end{split}$
Title stochastic integration by parts StochasticIntegrationByParts 2013-03-22 18:41:35 2013-03-22 18:41:35 gel (22282) gel (22282) 4 gel (22282) Theorem msc 60H05 msc 60G07 msc 60H10