A major difference between standard integral calculus and stochastic calculus is the existence of *quadratic variations* and *covariations*. Such terms show up, for example, in the stochastic version of the integration by parts formula.

For motivation, let us start by considering a standard argument for differentiable processes. The increment of a process over a time step can be written as . The following identity is easily verified,

(1) |

Now, divide the time interval into equal parts. That is, set for . Then, using and summing equation (1) over these times,

(2) |

If the processes are continuously differentiable, then the final term on the right hand side is a sum of terms, each of order , and therefore is of order . This vanishes in the limit , leading to the integration by parts formula

Now, suppose that are standard Brownian motions. Then, are normal random variables with standard deviation . It follows that the final term on the right hand side of (2) is a sum of terms each of which is, on average, of order . So, even in the limit as goes to infinity, it does not vanish. Consequently, in stochastic calculus, the integration by parts formula requires an additional term, which is called the *quadratic covariation* (or, just covariation) of and . Continue reading “Quadratic Variations and Integration by Parts”