The previous post introduced the idea of a purely discontinuous local martingale. In the context of that post, such processes were used to construct local martingales with prescribed jumps, and enabled us to obtain uniqueness in the constructions given there. However, purely discontinuous local martingales are a very useful concept more generally in martingale and semimartingale theory, so I will go into more detail about such processes now. To start, we restate the definition from the previous post.
Definition 1 A local martingale X is said to be purely discontinuous iff XM is a local martingale for all continuous local martingales M.
We can show that every local martingale decomposes uniquely into continuous and purely discontinuous parts. Continuous local martingales are well understood — for instance, they can always be realized as time-changed Brownian motions. On the other hand, as we will see in a moment, purely discontinuous local martingales can be realized as limits of FV processes, and arguments involving FV local martingales can often to be extended to the purely discontinuous case. So, decomposition (1) below is useful as it allows arguments involving continuous-time local martingales to be broken down into different approaches involving their continuous and purely discontinuous parts. As always, two processes are considered to be equal if they are equivalent up to evanescence.
Theorem 2 Every local martingale X decomposes uniquely as
where is a continuous local martingale with and is a purely discontinuous local martingale.
Proof: As the process is, by definition, equal to the jump process of a local martingale then it satisfies the hypothesis of Theorem 5 of the previous post. So, there exists a purely discontinuous local martingale with . We can take so that is a continuous local martingale starting from 0.
If is another such decomposition, then and have the same jumps and initial value so, by Lemma 3 of the previous post, . ⬜
Throughout the remainder of this post, the notation and will be used to denote the continuous and purely discontinuous parts of a local martingale X, as given by decomposition (1). Using the notation , and respectively for the spaces of local martingales, continuous local martingales starting from zero and the purely discontinuous local martingales, Theorem 2 can be expressed succinctly as
That is, is the direct sum of and . Definition 2 identifies the purely discontinuous local martingales to be, in a sense, orthogonal to the continuous local martingales. Then, (2) can be understood as the decomposition of into the direct sum of the closed subspace and its orthogonal complement. This does in fact give an alternative, elementary, and commonly used, method of proving decomposition (1). As we have already shown the rather strong result of Theorem 5 from the previous post, the quickest way of proving the decomposition was to simply apply this result. I’ll give more details on the more elementary approach further below.
Definition 1 used above for the class of purely discontinuous local martingales was very convenient for our purposes, as it leads immediately to the proof of Theorem 2. However, there are many alternative characterizations of such processes. For example, they are precisely the processes which are limits of FV local martingales in a strong enough sense. They can also be characterized in terms of their quadratic variations and covariations. Recall that the quadratic variation and covariation are FV processes with jumps and , so that they can be decomposed into continuous and pure jump components,
The following theorem gives several alternative characterizations of the class of purely discontinuous local martingales.
Theorem 3 For a local martingale X, the following are equivalent.
- X is purely discontinuous.
- for all continuous local martingales Y.
- for all local martingales Y.
- there exists a sequence of FV local martingales such that
Proof of 1 ⇒ 5: Setting , in the previous post we constructed a purely discontinuous local martingale satisfying via a sequence of FV local martingales with
Without loss of generality, we can set . Then, has the same initial value and the same jumps as X. So, . ⬜
Proof of 5 ⇒ 4: For any FV process Y we have the identity
Letting be FV local martingales as in statement 5,
The second inequality is a consequence of the Ito isometry. So, as required. ⬜
Proof of 4 ⇒ 3: If then the Cauchy-Schwarz inequality gives
Proof of 3 ⇒ 2: For any continuous local martingale Y, is continuous, so . ⬜
Proof of 2 ⇒ 1: If Y is a continuous local martingale, then is a local martingale and, hence, X is purely discontinuous. ⬜
In the previous post, it was shown that FV local martingales are purely discontinuous. In the opposite direction, we can say exactly when a purely discontinuous local martingale is an FV process.
Lemma 4 A purely discontinuous local martingale X is an FV process if and only if is almost surely finite for each .
Proof: For any cadlag process X, the sum is bounded by the variation of X over the interval and hence, is almost surely finite if X is an FV process.
Conversely, if is almost surely finite, we can define the process . This is an FV process with . If X is a local martingale then it is locally integrable and, hence, so is V. So, letting A be the compensator of V, its jumps satisfy
Common cases of purely discontinuous local martingales which are not FV processes include some Lévy processes. For a real-valued Lévy process X with charactistics then, as previously shown, it is locally integrable if and only if is finite, in which case it is integrable. Furthermore, using the notations of these notes, it is a martingale iff
For example, the purely discontinuous Lévy process with jump measure
and is an example of a purely discontinuous local martingale with infinite variation over all non-trivial intervals. Further examples are given by the symmetric stable Lévy processes with exponents , which have Lévy measure
The Local Martingale Decomposition
I will now look at the properties of decomposition (1). First, the continuous part of the quadratic covariation is the quadratic covariation of the continuous parts.
Lemma 5 If X and Y are local martingales then
Proof: From bilinearity of covariations,
The second equality here is using property 3 of Theorem 3, and the final equality uses the fact that the quadratic covariation of continuous processes is continuous. ⬜
Some approaches, (4) is used as the definition of the quadratic covariation. That is, for continuous local martingales, is the unique continuous FV process starting from zero such that is a local martingale. Then, for non-continuous local martingales, (4) defines the covariation of the continuous parts and, from (3), the covariation of X and Y is given by
Lemma 6 If X is locally an -integrable martingale, any , then so are and .
Proof: As is continuous, it is locally bounded and hence, is locally -integrable. Then, is locally -integrable whenever X is. ⬜
Showing that the decomposition preserves square integrable martingales is also straightforward.
Lemma 7 If X is a cadlag square integrable martingale then so are and and,
Proof: By property 2 of Theorem 3, and, hence,
The case of -integrability for is rather more complicated. Looking at the maximum process
Lemma 8 For each there exists a constant such that, for all local martingales X and times t,
Proof: Using from property 3 of Theorem 3,
So , with or . The Burkholder-Davis-Gundy inequality says that there are positive constants such that
for all local martingales M. Hence,
and the result holds with . ⬜
Lemma 9 For any , if X is an -integrable cadlag martingale then so are and . Furthermore, there exists a constant , independently of the choice of martingale X, such that
Proof: Setting or then, with as in Lemma 8,
The second inequality here is Doob’s martingale inequality. In particular, is integrable, so is of class (DL) and is a proper martingale}. Using the trivial bound , replacing by in (6) gives the result. ⬜
An obvious question to ask now, is whether Lemma 9 holds when p is equal to 1. I will not do so now, but it is possible to construct cadlag martingales whose continuous and purely discontinuous parts are not integrable and, therefore, are not proper martingales. So, Lemmas 8 and 9 are as far as we go in showing when the decomposition preserves integrability and the martingale property.
Above, we gave a very short proof of the local martingale decomposition in Theorem 2. This was possible because it follows quickly from the results of the previous post on constructing local martingales with prescribed jumps. Looking in more depth at the theory required, the constructions of the previous post required the theory of compensators. So, a full understanding of the proof of Theorem 2 does require some relatively advanced stochastic calculus theory. However, the decomposition is actually rather straightforward and can be understood — at least, for locally square integrable martingales — as nothing much more than orthogonal projection in Hilbert space. Decomposition (1) could have been stated and proven in these notes immediately following the definition of local martingales without requiring any of the development of stochastic integration or of the general theory. I did consider using such an elementary proof in the main body of the post above. However, the theory of compensators and the results of the previous post would still be required to describe the properties of purely discontinuous local martingales, such as in Theorem 3, so it would have been of limited benefit. Instead, I will now give an outline of how Theorem 2 can be proved with elementary properties of local martingales, and which is also closer to the approach used in many texts on the subject.
Use to denote the space of martingales X which can be expressed as
for a square integrable random variable . Replacing by if necessary, we can suppose that is -measurable. Martingale convergence tells us that is equal to the space of -bounded martingales and that, almost surely, . Although martingale convergence is not vital to the proof given here, it does help with the notation. We identify two elements of if, at each time, they are almost surely equal. This is weaker than equivalence up to evanescence, although it is the same if cadlag modifications are used.
The space of random variables can be identified with the space of by setting . The standard inner product in can be applied to ,
So, together with this inner product is a Hilbert space. The -bounded purely discontinuous martingales can be identified as the orthogonal complement of the set of continuous martingales starting from zero, which we denote by . That is, consists of the set of continuous with .
Lemma 10 A cadlag martingale is purely discontinuous if and only if for all .
Proof: First, if X is purely discontinuous then XY is a local martingale for all . As and are class (D), the same is true for . So, XY is a proper martingale and, by uniform integrability and the martingale property,
Conversely, suppose that for all Y in . Then, for any continuous , time and , the process
is a continuous martingale starting from zero with , so
So and XY is a martingale. By localization, this implies that XY is a local martingale for all continuous local martingales Y. ⬜
Now, for -bounded martingales, decomposition (1) reduces to an orthogonal projection.
Theorem 11 Every decomposes uniquely as
where and is purely discontinuous.
Proof: This is just orthogonal projection of X into the subspace , for which it just needs to be shown that is closed. To prove this, consider any sequence converging to a limit X in . Then, as previously shown, is continuous}. ⬜
So far, we have shown how to prove decomposition (1) for square integrable martingales using only basic properties of martingales. A straightforward localization procedure extends this to all locally square integrable martingales. However, not all martingales are locally square integrable, so it still remains to be shown that the result extends to general local martingales. In many approaches, a decomposition such as the following is used.
Lemma 12 Every local martingale can be decomposed as the sum of a locally bounded martingale and an FV local martingale.
Proof: Recall from the Bichteler-Dellacherie theorem that every semimartingale is the sum of a locally bounded martingale and an FV process. Alternatively, if X is a local martingale then, has locally integrable variation and is locally bounded. Then, letting A be the compensator of V, A is locally bounded and is an FV local martingale. So
decomposes X as the sum of a locally bounded martingale and an FV local martingale. ⬜
As FV local martingales are purely discontinuous and locally bounded processes are trivially locally square integrable, Lemma 12 allows decomposition (1) to be extended from the locally square integrable martingales to all local martingales. However, Lemma 12 did require either the Bichteler-Dellacherie theorem as stated in these notes, or the theory of compensators. So, it cannot be said to be elementary.
A more elementary approach is to instead approximate local martingales by square-integrable ones. By localization, we need only consider martingales of the form
for some integrable random variable . Then, choose a sequence, , of square integrable random variables such that tends to zero. Setting , we can apply the decomposition for square integrable martingales,
To complete the proof, it just needs to be shown that this converges to the decomposition for X. For that, it is enough to show that converges in the ucp topology. In fact, such convergence does hold as shown by the following result, which is related to Doob’s martingale inequality.
Lemma 13 If X is a cadlag martingale, then the continuous part satisfies
Proof: Define the stopping time
As the stopped process is a uniformly bounded local martingale, the product will be a local martingale and, as it is bounded by , it is of class (DL) and is a proper martingale starting from zero. So, by optional sampling it has zero expectation at time giving,
Next, by continuity, we have and whenever so,
The first inequality here is Markov’s inequality. ⬜
This concludes the outline of the alternative proof of Theorem 2. Although it is longer than that given further above, note that we did not make use of any stochastic calculus theory beyond the basic theory of filtrations and processes, and properties of spaces. The main idea is the orthogonal projection used in Theorem 11.