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 .
There are several methods of defining quadratic variations, but my preferred approach is to use limits along partitions. A stochastic partition of the nonnegative real numbers is taken to mean a sequence of stopping times, starting at zero and increasing to infinity in the limit,
Along such a partition, the approximation to the quadratic variation of and the approximation to the covariation of and is,
Note that the times are eventually constant and equal to , so the sums above only contain finitely many nonzero terms. If is a stochastic partition of then is a partition of the interval . Its mesh is the longest subinterval
For stochastic partitions, this is a random variable. The quadratic variation and covariation are then equal to the limit of the approximations (4) as the mesh goes to zero. At any fixed time, this limit converges in probability. However, looking at the paths of the processes, the stronger property of uniform convergence on compacts in probability (ucp) is obtained. The proof is given further below.
Theorem 1 (Quadratic Variations and Covariations) Let be semimartingales. Then, there exist cadlag adapted processes and satisfying the following.
For any sequence of stochastic partitions of such that, for each , the mesh tends to zero in probability as , the following limits hold
as . Furthermore, convergence also holds in the semimartingale topology.
The process is called the quadratic variation of and is the (quadratic) covariation of and . Note that, as is symmetric and bilinear in with , then the same holds for . That is,
for semimartingales and real numbers . It is clear that . The following properties also follow easily from the definition. Recall that an FV process is a cadlag adapted process with almost surely finite variation over all bounded time intervals, and such processes are semimartingales.
- If is a semimartingale then is a cadlag adapted and increasing processes.
- If are semimartingales then is an FV process.
Proof: Choose times . If is any stochastic partition as in (3) with for some then,
Taking limits of such partitions gives as required. Then, the polarization identity expresses as the difference of increasing processes and, as such, is an FV process. ⬜
In particular, quadratic variations variations and covariations are semimartingales. The following differential notation is sometimes used,
The stochastic version of the integration by parts formula is as follows. Sometimes, this result is used as the definition of quadratic covariations instead of using partitions as above.
Theorem 3 (Integration by Parts) If are semimartingales then
The proof of this is given along with the existence of quadratic variations below. The special case with is often useful,
Note that, in differential notation (6) is simply
which is the stochastic differential version of equation (1). As is a semimartingale, the following corollary of the integration by parts formula is obtained.
Corollary 4 If are semimartingales, then so is .
Quadratic Variation of Brownian motion
As standard Brownian motion, , is a semimartingale, Theorem 1 guarantees the existence of the quadratic variation. To calculate , any sequence of partitions whose mesh goes to zero can be used. For each , the quadratic variation on a partition of equally spaced subintervals of is
The terms are normal with zero mean and variance . So, their squares have mean and variance , giving the following for ,
The variance vanishes as goes to infinity, . This gives the quadratic variation as simply
Using bilinearity of quadratic covariations, this result can be generalized to obtain the quadratic covariations of correlated Brownian motions. Two Brownian motions have correlation if, for each , and are independent of and jointly normal with correlation . That is, their covariance is . If for a constant then it follows that is normal with variance
If we choose then this shows that is a standard Brownian motion, with quadratic variation given by (8). Bilinearity of covariations can be applied,
Substituting back in , rearranging this expression gives the following for the covariation of the correlated Brownian motions,
In particular, independent Brownian motions have zero covariation.
Existence of Quadratic Variations
It remains to prove that the quadratic variations and covariations defined by Theorem 1 do indeed exist, and then that the integration by parts formula is satisfied. In fact, it is easier to first define to be the unique process satisfying equation (6), and then show that the limit stated by Theorem 1 holds. It follows directly from this definition that is a cadlag adapted process. Taking then it is enough to prove the limit . The corresponding limit for the covariation will follow from the polarization identities and .
The method of proof will be to express as a stochastic integral, so that the dominated convergence theorem can be used to show that this tends to zero as the mesh of the partition vanishes. Fixing a time and partition , as in equation (3), the square of the change in across an interval, , can be rearranged as
The last term on the right can be expressed as the integral of with respect to between the limits and . Also substituting in the integration by parts formula (7) for the first two terms,
If we sum this expression over , then the integrand in the final term becomes
giving the following expression for the quadratic variation along a partition
Now, let be a sequence of such partitions whose mesh goes to zero as . It is clear from the left continuity of that . Furthermore, , which is locally bounded. Then, the dominated convergence theorem says that , converging ucp and in the semimartingale topology. Putting this into (9) gives as required.
This proves the result in the case where everywhere for all . The final thing to do is to generalize to the case where only goes to zero in probability. However, in that case it is possible to pass to a subsequence satisfying for all . The Borel-Cantelli lemma guarantees that almost everywhere for all . The above proof then shows that , converging ucp and in the semimartingale topology. Then, by this same argument, every subsequence of itself has a subsequence converging to . As the cadlag processes under the ucp and semimartingale topologies is a metric space, this is enough to guarantee convergence of the original sequence.