Special classes of processes, such as martingales, are very important to the study of stochastic calculus. In many cases, however, processes under consideration `almost’ satisfy the martingale property, but are not actually martingales. This occurs, for example, when taking limits or stochastic integrals with respect to martingales. It is necessary to generalize the martingale concept to that of local martingales. More generally, localization is a method of extending a given property to a larger class of processes. In this post I mention a few definitions and simple results concerning localization, and look more closely at local martingales in the next post.
Definition 1 Let P be a class of stochastic processes. Then, a process X is locally in P if there exists a sequence of stopping times such that the stopped processes
are in P. The sequence is called a localizing sequence for X (w.r.t. P).
I write for the processes locally in P. Choosing the sequence of stopping times shows that . A class of processes is said to be stable if is in P whenever X is, for all stopping times . For example, the optional stopping theorem shows that the classes of cadlag martingales, cadlag submartingales and cadlag supermartingales are all stable.
Definition 2 A process is a
- a local martingale if it is locally in the class of cadlag martingales.
- a local submartingale if it is locally in the class of cadlag submartingales.
- a local supermartingale if it is locally in the class of cadlag supermartingales.
The class of cadlag martingales is denoted by , and the class of local martingales is written as . Furthermore, for , a martingale X is said to be -integrable if is -integrable for each time. That is . Then, denotes the cadlag -integrable martingales, and the local -integrable and cadlag martingales.
Another property which is frequently useful in a local form is that of uniformly bounded processes. A process X is uniformly bounded if, almost surely, for all times t and some constant K.
Definition 3 A process is locally bounded if it is locally in the class of uniformly bounded processes.
As an example, all continuous adapted processes are locally bounded, with the localizing sequence . More generally, this holds for left-continuous adapted processes which are almost surely bounded on every finite time interval, which includes all left-continuous adapted process with right limits.
A process X is integrable if is integrable at each time t. Furthermore, for any , it is integrable if is -integrable at all times. For this means that is finite, and for it means that is uniformly bounded. Unfortunately, these definitions do not give stable classes. Instead, I define local integrability as follows. Recall that, for a process X, its maximum process is .
Definition 4 A process X is locally integrable if is locally in the class of integrable processes.
More generally, for any , X is locally -integrable if is locally in the class of -integrable processes.
Equivalently, for , the process X is locally -integrable iff is locally integrable. Similarly, it is locally -integrable iff it is locally bounded. As the class of processes whose maximum process is -integrable is stable, they behave well under localization and the definition of local integrability given above works well. Also, for nonnegative increasing processes, , in which case it is not necessary to refer to the maximum process in the definition above. The textbook definition of local integrability is often only applied to increasing processes, and the definition I state above is a useful generalization to arbitrary processes.
Localizing a pair of properties separately is equivalent to localizing the combination of the properties. For example, is equal to the space of processes which are both a local martingale and also locally -integrable.
Lemma 5 If P,Q are stable classes of processes then
Proof: The inclusion is trivial. Conversely, suppose that the process X is in . Let be localizing sequences with respect to P and Q respectively. By stability of P,
So, . Similarly, . ⬜
Localizing a property is something which only needs to be done at most once, as repeated localization has no effect, as stated in the following result. For example, the space of processes which are locally in is just the same thing as the space of local martingales.
Lemma 6 If P is a stable vector space of processes then .
Proof: If then there is a sequence of stopping times such that Then, there are sequences of stopping times increasing to infinity as for each fixed , and . Setting gives a countable set of stopping times with , and by the following lemma. ⬜
Lemma 7 Let P be a stable vector space of processes. Then, a process X is locally in P if and only if there is a sequence of stopping times with and such that .
Proof: If X is locally in P, then any localizing sequence satisfies the required properties. Conversely, suppose that satisfy the required properties. Then, are stopping times increasing to infinity. It just needs to be shown that are in P. Using induction, suppose that this is true for a given n.
By the induction hypothesis and stability of P, each of the terms on the right hand side is in P and, as this is a vector space, so is the left hand side. Therefore, is a localizing sequence. ⬜
As was noted above, continuous and adapted processes are always locally bounded and, hence, locally integrable. More generally, for cadlag adapted processes, local integrability can be described in terms of the jumps of the process.
Lemma 8 For any then a cadlag adapted process X is locally -integrable if and only if is locally -integrable.
Proof: Note that is -integrable whenever is. Therefore, is locally -integrable whenever X is.
Conversely, suppose that is -integrable. Defining the stopping times gives
whenever , which is integrable. Then, is a localizing sequence showing that X is locally -integrable. So, applying Lemma 6, X is locally -integrable whenever is. ⬜
The class (D) property is easily seen to be stable, and should localize nicely. However, this just leads to local integrability.
Lemma 9 For a cadlag adapted process X, the following are equivalent.
- X is locally integrable.
- X is locally of class (DL).
- X is locally of class (D).
Proof: First, if is integrable then, for each time , the set of random variables for stopping times is dominated by the integrable variable , and hence is uniformly integrable. So, X is of class (DL). Localizing, all locally integrable processes are locally of class (DL).
Any process X of class (DL) is locally of class (D), using the localizing sequence .
Now, suppose that X is cadlag, adapted, and of class (D). Setting gives
which, by integrability of , is integrable. So, X is locally integrable and, applying Lemma 6, this still holds whenever X is locally of class (D). ⬜
It is useful to know that local martingales, submartingales and supermartingales are locally integrable.
Lemma 10 Every local martingale, local submartingale and local supermartingale is locally integrable.
Proof: Let X be a local martingale, submartingale or supermartingale. By stability of the local integrability property, it is enough to show that X is locally a locally integrable process. So, we can suppose that X is a proper submartingale or supermartingale. Define the stopping times
for each positive integer n, and set . These times increase to infinity as n goes to infinity and inequality (1) holds. So, it just needs to be shown that is integrable. However, as are bounded stopping times, this is stated by optional sampling. ⬜
It is frequently useful to be able to take conditional expectations of a process at a stopping time. In general, for this to be well-defined requires the process to satisfy some integrability properties. The following lemma shows that local integrability of the process is sufficient.
Lemma 11 If X is locally -integrable then, for any stopping time , the conditional expectations
are almost surely-finite.
Proof: By local integrability, there exist stopping times increasing to infinity such that is -integrable for all n. As and are -measurable,
As n goes to infinity, we have for large enough n whenever . So, the conditional expectation on the left hand side is almost-surely finite when . Exactly the same argument holds with in place of . ⬜
Finally, I will mention that sometimes it is useful to localize a process by stopping just before stopping times , rather than at those times. This is called prelocalization, and can be useful to avoid sudden jumps in the process at inaccessible times. I do not make much use of prelocalization in these notes, but will now briefly look at prelocally integrability. Compare the following with Definition 4 above. Here, is used to denote the left limits of the process ,
and we take to be 0.
Definition 12 A process X is prelocally integrable if is locally in the class of integrable processes.
More generally, for any , X is prelocally -integrable if is locally in the class of -integrable processes.
As , it should be clear that prelocal integrability is a weaker property than local integrability. In fact, all cadlag adapted processes are prelocally integrable.
Lemma 13 If X is a right-continuous adapted process such that, for each time t, is almost surely finite, then X is prelocally -integrable.
In particular, every cadlag adapted process is prelocally -integrable.
Proof: Define the stopping times
As for n greater than , the sequence increases to infinity under the hypothesis of the lemma. Also, is bounded by n, so is -integrable, and X is prelocally -integrable.
As is finite for any cadlag process X, cadlag adapted processes are prelocally -integrable. ⬜
Finally, Lemma 11 extends to prelocally integrable processes, although the conclusion is only of any use if X is not a progressively measurable process (e.g., if it is not adapted).
Lemma 14 If X is prelocally -integrable then, for any stopping time , the conditional expectation
is almost surely-finite.
Proof: The proof is almost identical to that above for Lemma 11. By prelocal integrability, there exist stopping times increasing to infinity such that is -integrable for all n. As is -measurable,
As n goes to infinity, we have for large enough n whenever . So, the conditional expectation on the left hand side is almost-surely finite when . ⬜