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