The previous two posts introduced the concept of quasimartingales, and noted that they can be considered as a generalization of submartingales and supermartingales. In this post we prove various basic properties of quasimartingales and of the mean variation, extending results of martingale theory to this situation.
We start with a version of optional stopping which applies for quasimartingales. For now, we just consider simple stopping times, which are stopping times taking values in a finite subset of the nonnegative extended reals . Stopping a process can only decrease its mean variation (recall the alternative definitions and for the mean variation). For example, a process X is a martingale if and only if , so in this case the following result says that stopped martingales are martingales.
Lemma 1 Let X be an adapted process and be a simple stopping time. Then
Assuming, furthermore, that X is integrable,
and, more precisely,
where the supremum is taken over elementary processes . Choosing any elementary processes and , set . As is a simple stopping time, this is elementary and,
Taking the supremum over all such ,
The supremum is over all elementary processes with time index running over . Let us also extend the stopped process to this index set, by setting . Then,
as required. ⬜
This result can be used to extend Doob’s submartingale inequality to quasimartingales.
Theorem 2 Let X be a cadlag adapted process and K be a nonnegative real. Then,
Proof: For any finite , define the simple stopping time
Then, and whenever . So, letting ,
The final inequality here is an application of lemma 1. Now, let be a countable dense subset of and be a sequence of finite subsets of increasing to . By right-continuity, if then, for any , we have for large . So,
Letting increase to gives the result. ⬜
Next, we look at stochastic integrals with respect to elementary integrands. From the mean variation of a process, it is not possible to give -bounds for stochastic integrals with respect to that process, even for uniformly bounded integrands. However, the integrals are bounded in probability. The following generalises Lemma 4 of the post on martingales as integrators. In particular, from the post on existence of stochastic integrals, this implies that quasimartingales are semimartingales, although this fact also follows from Rao’s theorem decomposition and the fact that submartingales are semimartingales.
Lemma 3 There exists a constant such that, for any elementary predictable process , adapted process X, and ,
Proof: I will prove this by extending Lemma 4 from the post on martingales as integrators, which says that there is a such that
for any martingale .
We suppose that is finite so that X is integrable, otherwise the result is trivial. Now, there exists a sequence of times such that
where is -measurable. Now consider a Doob decomposition
where , and is a martingale restricted to the index set . We write for the discrete integral . The definition of gives
Similarly, we can bound the martingale M in ,
Hence, as stated above,
Combining these inequalities,
So the result follows by replacing by in the above. ⬜
Next, the mean variation behaves as we would expect under limits.
Lemma 4 Let be integrable adapted processes such that in as n goes to infinity. Then,
Proof: Letting be an elementary process on time index set , convergence gives
Taking the supremum over all such gives the inequality for . The inequality for follows in the same way, except that we set and let range over the elementary processes with time index set . ⬜
The optional stopping result, Lemma 1, can be extended to arbitrary stopping times. The proof will require approximating by simple stopping times and taking limits in but, before doing this, we will need to know that such approximations converge in . To that end, we extend lemma 6 of the post on cadlag modifications to the quasimartingale case. This requires a filtered probability space for an arbitrary linearly ordered time index set . For an integrable process defined on such a space, we define the mean variation as
The supremum is taken over all finite sequences in . Then, the following lemma holds.
Lemma 5 Let X be an integrable adapted process with respect to a filtered probability space such that .
Then, is uniformly integrable for any decreasing sequence .
Proof: Define the random variable
Monotone convergence shows that this is integrable and, in particular, is almost surely finite
So, the following sum is almost surely convergent, and converges in
Furthermore, , so the sequence is uniformly integrable. Then, from the definition of , the martingale property
is satisfied and the sequence is uniformly integrable. So, is a uniformly integrable sequence. ⬜
We now extend optional stopping to arbitrary stopping times.
Lemma 6 Let X be a right-continuous adapted process and be a stopping time. Then
Assuming, furthermore, that X is a quasimartingale, then is integrable and
and, more precisely,
Proof: For the first inequality we can assume without loss of generality that is finite, so that X is integrable. So, in either case, we can suppose that X is integrable and is finite.
Now, choose a sequence of simple stopping times decreasing to as n goes to infinity. We show that in , To do this, let be the negative integers and set and for each . We can show that is finite. For any decreasing sequence of positive integers ,
where is the elementary process
Hence the previous equation has expectation bounded by and, so, . Lemma 5 then says that the sequence is uniformly integrable so, by right-continuity of X, converges to in , as required.
giving (4). Similarly,
An immediate consequence of this is that the class of quasimartingales is stable with respect to localization.
Lemma 7 If X is a cadlag quasimartingale, then so is for any stopping time
Proof: Applying lemma 6,
So is a quasimartingale. ⬜
Lemma 8 A cadlag process X is locally a quasimartingale if and only if it is a locally integrable semimartingale.
Proof: First, if X is a cadlag quasimartingale then we can define stopping times
The stopped process is a quasimartingale and, hence, integrable. Then, is integrable. Hence, X is locally integrable. Therefore, if X is locally a quasimartingale then it is locally integrable. Furthermore, we have noted above (see Lemma 3) that every quasimartingale is a semimartingale, so X is also a semimartingale.
Conversely, suppose that X is a locally integrable semimartingale. As shown previously, this means that we can decompose
for a local martingale M and predictable FV process A. Letting be the variation of A on the interval , then is predictable and, hence, V is locally bounded. Then, we can find stopping times increasing to infinity such that is a martingale, and has uniformly bounded variation. For any elementary ,
So, is finite, and is a quasimartingale. ⬜
The martingale convergence theorem also extends to quasimartingales, as we show now.
Theorem 9 Let X be a cadlag and integrable adapted process with . Then, almost surely, the limit exists and is finite.
Proof: It is possible to prove this in the same way as for martingale convergence by using quasimartingales from the start. Here, however, I will leverage the martingale convergence result, as we have already proved this. Rao’s decomposition shows that for nonnegative cadlag supermartingales Y, Z. As previously shown, converges almost surely as t goes to infinity, so also converges. ⬜
As was noted in the initial post on quasimartingales, the mean variation is bounded by the expected value of the pathwise variation. However, in general we just get an upper bound and not equality. As we now show, equality is attained in the case of predictable FV processes.
Lemma 10 If X is an integrable and predictable FV process then,
To start with, we will suppose that X has integrable variation on . There exists a predictable process with
Now, by an application of the monotone class theorem, the elementary predictable processes are dense in the predictable processes, in the sense that we can find elementary such that
as . Replacing by will just decrease the left hand side of the above limit, so we can assume that . Using dominated convergence and the integral definition for ,
This gives (8) in the case that X has integrable variation. More generally, the variation of X will be locally bounded, so we can find stopping times increasing to infinity such that the stopped process has integrable variation. Applying Lemma 6,
The limit here is taking and using monotone convergence. This proves (8). ⬜
Finally for this post, we show that mean variation is well behaved under taking limits in probability, rather than the much stronger convergence in of Lemma 4. Although the following result seems simple, and is reminiscent of Fatou’s lemma, it is rather stronger than it might seem at first sight. For example, it does not hold if is replaced by . It is not difficult to construct an -bounded sequence of martingales (hence ) which converges in probability to a non-martingale , so . Such examples are given by stopping local martingales which are not martingales. The following result does however show that the limit of such a sequence is a quasimartingale — see corollary 12 below.
Theorem 11 Let be a sequence of adapted processes such that, for each , in probability as n goes to infinity. Then,
Before proceeding with the proof of this theorem, I will first note an alternative method of proof which gives the result quickly in an intuitive way, although making it rigorous involves more work. We may assume that the right hand side of (9) is finite, otherwise the result is trivial. Then, the statement is unchanged if we restrict to a subsequence for which is finite.
Rao’s theorem says that we can decompose each as the difference, , of nonnegative supermartingales for which . If the sequences converge in probability to limits Y,Z, then . Fatou’s lemma can be used to show that Y,Z are nonnegative supermartingales and,
A difficulty with this approach is that need not converge to a limit, although it is possible to enforce this property by passing to convex combinations of the sequence and using Komlós’s subsequence theorem (J. Komlós: A Generalization of a Problem of Steinhaus, Acta Math. Hung. 18 (1967), 217-229.)
The proof of theorem 11 which I give now is a bit longer, but does not require any results such as Komlós’s theorem or Rao’s decomposition. Instead, it just uses several applications of Fatou’s lemma.
Proof: Setting then, as shown previously in the post on quasimartingales, the mean variation can be expressed as
However, Fatou’s lemma requires non-negative integrands, which is not the case here, and the first inequality above does hold in general.
Instead, we will proceed by breaking down the left hand side into non-negative terms to which Fatou’s lemma does apply. First, we can restrict to the case where the right hand side of (9) is finite, otherwise the result is trivial. Then, we restrict to the subsequence where , as this does not affect the statement of the result.
For any t, Fatou’s lemma gives
So, X is integrable.
Now, for any and -measurable random variable , the terms are non-negative, so Fatou’s lemma can be applied to this
Subtracting from both sides,
Jensen’s inequality shows that the final term on the right hand side is nonnegative
The final inequality here is just the triangle inequality. So, we can take expectations of (10) and again apply Fatou’s lemma
Now, the elementary process can be expressed as
for all , where are fixed times and are -measurable random variables. So,
as required. ⬜
As was mentioned above, this result shows that limits of -bounded sequences of martingales are quasimartingales. Noting that martingales satisfy and, hence, ,
Corollary 12 Let () be a sequence of martingales, such that in probability as goes to infinity. Then
In particular, if is cadlag and is bounded for each , then is a quasimartingale.
Proof: By theorem 11 applied to the processes stopped at time t, together with the martingale property,
as required. ⬜