Being able to handle quadratic variations and covariations of processes is very important in stochastic calculus. Apart from appearing in the integration by parts formula, they are required for the stochastic change of variables formula, known as Ito’s lemma, which will be the subject of the next post. Quadratic covariations satisfy several simple relations which make them easy to handle, especially in conjunction with the stochastic integral.
Recall from the previous post that the covariation is a cadlag adapted process, so that its jumps are well defined.
Lemma 1 If are semimartingales then
In particular, .
Proof: Taking the jumps of the integration by parts formula for gives
and rearranging this gives the result. ⬜
An immediate consequence is that quadratic variations and covariations involving continuous processes are continuous. Another consequence is that the sum of the squares of the jumps of a semimartingale over any bounded interval must be finite.
Corollary 2 Every semimartingale satisfies
Proof: As is increasing, the inequality holds. Substituting in gives the result. ⬜
Next, the following result shows that covariations involving continuous finite variation processes are zero. As Lebesgue-Stieltjes integration is only defined for finite variation processes, this shows why quadratic variations do not play an important role in standard calculus. For noncontinuous finite variation processes, the covariation must have jumps satisfying (1), so will generally be nonzero. In this case, the covariation is just given by the sum over these jumps. Integration with respect to any FV process can be defined as the Lebesgue-Stieltjes integral on the sample paths, which is well defined for locally bounded measurable integrands and, when the integrand is predictable, agrees with the stochastic integral.
Lemma 3 Let be a semimartingale and be an FV process. Their covariation is
In particular, if either of or is continuous then .
Proof: Expressing the covariation as the limit along equally spaced partitions of gives
The third equality here makes use of the bounded convergence theorem to commute the limit with the integral sign. Then, as is cadlag, on each sample path there is only a countable set of times at which . Bounded convergence can again be used to evaluate the integral,
as required. ⬜
If are semimartingales and are continuous FV processes then,
That is, when calculating covariations, we can disregard any continuous FV terms added to the processes. A consequence of Lemma 3 is that the standard integration by parts formula, with no covariation term, applies whenever either of the two processes has finite variation. The integral with respect to the FV process in the following is the Lebesgue-Stieltjes integral on the sample paths. As need not be predictable, it is not always defined as a stochastic integral.
Corollary 4 Let be a semimartingale and be an FV process. Then,
Proof: Substitute (2) for the covariation term in the integration by parts formula to get the following
The final two integrals on the right hand side can be combined into a single integral of , giving the result. ⬜
A d-dimensional process is said to be a semimartingale if each of its components are semimartingales. The quadratic variation is defined as the dxd matrix-valued process
This will also be increasing, in the sense that is almost surely positive semidefinite for all times . That is,
is increasing for all vectors . Here, I am using the summation convention, where the indices appearing twice in a single term are summed over. This gives the following result for the integral with respect to .
Lemma 5 Let be semimartingales and be bounded and measurable processes. Then,
is an increasing process.
Proof: Almost surely, is an increasing matrix valued process, as mentioned above. Restricting to any fixed sample path satisfying this property, consider a process of the form
for and times . This has integral
which is increasing. The idea is to apply the functional monotone class theorem to extend this to all bounded and measurable processes. So, let be the set of bounded measurable functions for which (3) gives an increasing process. By the argument above, this includes all step functions of the form (4). By the monotone class theorem, to show that contains all bounded measurable functions it is enough to show that if is a uniformly bounded sequence tending to the limit , then . However, this follows from the bounded convergence theorem for the Lebesgue-Stieltjes integrals with respect to and the fact that a limit of increasing functions is increasing. ⬜
The quadratic covariation considered as a bilinear map is symmetric and positive semidefinite. The Cauchy-Schwarz inequality gives the following bound for the covariation,
More generally, the previous result can be used to obtain the Kunita-Watanabe inequality.
Theorem 6 (Kunita-Watanabe Inequality) Let be semimartingales and be measurable processes. Then,
Proof: First, suppose that are bounded. Considering the 2-dimensional semimartingale and for a fixed , the previous result says that
is an increasing process. As the first two terms are increasing, and the variation of the third term is this gives the following inequality,
The result follows by setting . Finally, this extends to unbounded integrands by monotone convergence. ⬜
For example, consider standard Brownian motions . These have quadratic variation and the Kunita-Watanabe inequality says that
The Radon-Nikodym theorem can then be used to imply the existence of a predictable process with . This is the instantaneous correlation of the Brownian motions. Recall from the previous post that this is consistent with the quadratic covariations where the correlation is a fixed number.
We now arrive at the following result allowing us to commute the order in which stochastic integrals and quadratic covariations are calculated. This is a very useful result which is often required when manipulating stochastic integrals. Note that equations (6) and (7) can be written in differential form as follows,
Theorem 7 Let be semimartingales and be an -integrable process. Then, is -integrable in the Lebesgue-Stieltjes sense,
Furthermore, the quadratic variation of is given by,
As an example, consider Ito processes ,
where are Brownian motions with correlation , so . As mentioned above, the continuous finite variation parts and do not contribute to the covariation. Theorem 7 gives,
The proof of Theorem 7 is as follows.
Proof: First, consider the case where is an elementary predictable process, so that it is left-continuous and piecewise constant, with discontinuities at some finite set of times . Setting then for any interval over which is constant. Then, the covariation calculated along a partition containing satisfies,
Taking the limit of such partitions gives (6).
Now, let be the set of processes such that (6) is satisfied. As just shown, this includes the elementary predictable processes. The idea is to apply the functional monotone class theorem to prove that .
Let be a sequence converging to a limit , and dominated by some . Setting , integration by parts gives
By the dominated convergence theorem, the first two terms on the right converge ucp to the limit with and in place of and . Furthermore, as , we may pass to a subsequence such that this almost surely converges uniformly on compacts. Then, the dominated convergence theorem shows that the final term on the right hand side of (8) also converges to the limit with in place of . Taking the limit and applying integration by parts a second time gives,
In particular, if is a uniformly bounded sequence then applying bounded convergence to the left hand side gives (6), so that . The monotone class theorem then says that contains all bounded predictable processes.
Next, suppose that are bounded predictable processes dominated by . Then, (9) says that and, by definition, is -integrable in the sense of stochastic integration. Therefore, .
So far, we have shown that every -integrable process is also -integrable, and equation (6) is satisfied.
Finally, equation (5) follows from the Kunita-Watanabe inequality,
A simple consequence of Theorem 7 is that stopping the covariation of two semimartingales at a stopping time is the same as stopping either of the individual processes.
Corollary 8 If are semimartingales and is a stopping time then,
In particular, .
Proof: The result follows by integrating ,
A semimartingale which is small in absolute value does not necessarily have a small quadratic variation or, to state this another way, quadratic variation is not a continuous map under ucp convergence. For example, consider solutions to the stochastic differential equation for a Brownian motion B, constant , and initial condition . This is an Ornstein-Uhlenbeck process with mean reversion rate . It can be shown that, as a function of , the solution X will converge ucp to zero in the limit as , but, the quadratic variation does not depend on at all. Continuity of quadratic variation and covariation can be recovered by, instead, using the stronger semimartingale topology. In the following, denotes the space of semimartingales and denotes convergence of a sequence to in the semimartingale topology.
Lemma 9 Quadratic covariation defines a jointly continuous map
under the semimartingale topology.
That is, if , , X, Y are semimartingales with and , then and, furthermore, the variation of on any bounded interval tends to zero in probability.
Proof: We start by showing that if then in probability for a fixed time t. As semimartingale convergence implies ucp convergence, the stopping times
tend to infinity in probability. So, it is enough to show that tends to zero in probability. Using integration by parts,
The first two terms on the right hand side tend to zero in probability, by ucp convergence. For the final term, for any fixed n, note that is left-continuous, adapted, and bounded by 1 and, hence, can be written as the limit of a sequence of elementary predictable processes . For example, we can take over . Recalling that denotes the supremum of over bounded elementary processes ,
Here, bounded convergence has been used to write the integral as a limit of integrals over . By definition of the semimartingale topology, as n tends to infinity and, therefore, we have shown that tends to zero in probability. So, in probability as claimed.
Now suppose that and . By the bilinearity of quadratic covariations,
Applying the Kunita-Watanabe inequality (Theorem 6) to each of the three terms on the right hand side shows that this has variation bounded by
on an interval . However, by what was shown above, this tends to zero in probability as n goes to infinity. So, we have proved the `furthermore’ part of the statement.
Finally, letting denote the variation of over the interval and letting be a sequence of elementary processes,
in probability as n tends to infinity. So tends to in the semimartingale topology. ⬜