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 1Ifare semimartingales then

(1)

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 2Every semimartingalesatisfies

*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 3Letbe a semimartingale andbe an FV process. Their covariation is

(2)

In particular, if either oforis 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 4Letbe a semimartingale andbe 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 5Letbe semimartingales andbe bounded and measurable processes. Then,

(3)

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

(4) |

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)Letbe semimartingales andbe 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 7Letbe semimartingales andbe an-integrable process. Then,is-integrable in the Lebesgue-Stieltjes sense,

(5)

and,

(6)

Furthermore, the quadratic variation ofis given by,

(7)

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

(8) |

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,

(9) |

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, .

Any -integrable process is the limit of a sequence of bounded predictable processes . As , dominated convergence implies that left hand side of (9) tends to , giving (6) as required.

So far, we have shown that every -integrable process is also -integrable, and equation (6) is satisfied.

Setting , we have shown that is -integrable and . Similarly, is -integrable satisfying . So, by associativity of stochastic integration, is -integrable and equation (7) follows,

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 8Ifare semimartingales andis 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 9Quadratic covariation defines a jointly continuous map

under the semimartingale topology.

That is, if,,X,Yare semimartingales withand, thenand, furthermore, the variation ofon 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. ⬜

Dear George,

I’m trying to prove a certain type of continuity of the quadratic variation for a fixed time T: for all \epsilon > 0, there exists \delta > 0 such that \|X\|_{+\infty} < \delta \Rightarrow [X]_T < \epsilon. Here \|X\|_{+\infty} = sup_{0 \leq t \leq T} |X_t|. Both inequalities should hold almost surely and X is just a bounded cadlag process in [0,T]. Do you know any reference for this? I've looked for it on a lot of books (Protter, Karatzas and Shreve,…), but I couldn't find anything.

Thank you very much and congratulation, the site is really awesome!

Yuri – There is no such relation between the maximum of a process and its quadratic variation. Actually, X can remain as close to zero as you like at the same time as its quadratic variation being as large as you like. I’ll post an example when I have time to log on.

Start with Brownian motion and make then it jump to zero every time |B_t|=delta > 0. It picks up q-variation at dt + jumps and is uniformly bounded by delta.

Gerard – Yes, that works. I was thinking of something like an Ornstein-Uhlenbeck process with high mean reversion rate. Or

f(W_{t}) for a standard Brownian motionWandfan arbitrarily small function with non-vanishing derivative. Something likef(x) =asin(a^{−1}x) will do.Hi Gerard and George, I have just seen your examples. Thank you very much, they were really helpful. Best.

I don’t think I understand exactly what Lemma 5 is about. So to move it in a easier case if I have Z=(X,Y) semimartingale – so X,Y semimartingales – and M bounded measurable process I have that is an increasing process?

I image that “increasing” here means with respect to the partial ordering on symmetric matrices, not that the individual components are increasing.

soumik – Yes, I am referring to the matrix rather than the individual components (as I mention above the statement of the Lemma). The summation convention is being used in equation (3), so in the case of a pair of semimartingales

X,Yit actually means thatis increasing for bounded measurable processes

M,N.Update: I have added an additional result to this post, Lemma 9, showing that quadratic variations and covariations are continuous under the semimartingale topology.This also relates to Yuri’s question in the comment above. Although a semimartingale which is small in absolute value need not have a small quadratic variation, it is true that a semimartingale which is close to zero in the semimartingale topology has a small quadratic variation (in probability).

Hello,

Thank you for all your posts.

For Lemma 9, is there an equivalent result for Skhorokhod topology instead of semimartingale topology?

Hi. No, I don’t think that there is much you can say here for the Skorokhod topology. Note that the Skorokhod topology is weaker than uniform convergence, so any of the examples given by me and Gerard above are also examples of processes whose paths tend to zero under the Skorokhod topology, but whose quadratic variation does not go to zero.

Dear George,

I’m wondering whether you have posted anywhere about the convergence of the quadratic covariation.

Say $X_n$ and $Y_n$ respectively converge to semimartingales $X$ and $Y$. It would be interesting to know what kind of convergence you would need to impose on $X_n$ and $Y_n$ for their to get convergence of the covariations to the covariation of the limiting semimartingales.

Great post once again!

Sorry, but no I haven’t looked at this beyond Lemma 9 above. Convergence in the semimartingale topology is sufficient. You could, no doubt, come up with weaker conditions which are still sufficent and relevant to whatever specific problems you are looking at. However, I don’t think that the other common kinds of convergence weaker than semimartingale convergence are sufficient on their own.

Thank you for this tremendous work you put in to make these easy to read note available online.

I have two questions though.

1- after equation 2 when you said ” of order….” its in the mean sqare sense right?

2- does the use random partition lead to a largely different quadratic variation concept compared to what we get when using deterministic partition ?

Thanks in advance

Your proof of the Kunita-Watanabe Inequality resembles that of the Cauchy-Schwarz inequality. I was wondering whether one could argue more directly: On the space of measurable processes paired with semimartingales define the inner product . Denote by the random (-by-) Hahn decomposition of . Conclude by applying C-S to .

I also meant to say thanks for the excellent blog!

How do i have to interpret the sum in Corollary 2? Is it possible for a semimartingale to have infinitely many jumps in the interval ?

Hi. No, it can be an infinite sum. There will be only countably many jumps but, generally, infinite sums of non-negative numbers are well-defined.

Hi, thank you for your answer. So the countability is because is a càdlàg process and therefore we have only countably many s s.t. the jump process of at time s is greater 1/n for each n (which was proven here: http://math.stackexchange.com/q/1914422/337225). Therefore there are only countably many s s.t. the jump process is greater than zero, since the countable union of countably many jumps is again countable. Is that correct?

Correct, although maybe it is a bit more direct to use the fact that X is cadlag rather than [X].

On the right side of your formula in the proof of lemma 1, there are probably two “dots” missing for the integration. Because i read

as normal multiplication. Applying the lemma gives the desired equality.

I did mean this as normal multiplication, and I don’t quite follow what you mean by .

The integration by parts formula tells us that . Assuming or , we have and thus . But now you probably apply (that was actually wrong in my first comment), which is only available for locally bounded predictable processes. But i guess and are locally bounded and predictable.

I did indeed apply that result. Corollary 8 from the post on properties of stochastic integral. Actually, locally bounded is not required, just Y-integrability. We do have local boundedness here though, which is how we can be sure that is Y integrable in the first place.

Dear George,

In the proof of equation (8) of Theorem 7 you mention that the sequence in the final term on the right hand side is dominated but you don’t mention which is the dominator. I suppose you can claim that is caglad and proceed.

Yes, the fact that it converges uniformly guarantees that the supremum is finite, and caglad.