The optional and predictable projections of stochastic processes have corresponding dual projections, which are the subject of this post. I will be concerned with their initial construction here, and show that they are well-defined. The study of their properties will be left until later. In the discrete time setting, the dual projections are relatively straightforward, and can be constructed by applying the optional and predictable projection to the increments of the process. In continuous time, we no longer have discrete time increments along which we can define the dual projections. In some sense, they can still be thought of as projections of the infinitesimal increments so that, for a process *A*, the increments of the dual projections and are determined from the increments of *A* as

(1) |

Unfortunately, these expressions are difficult to make sense of in general. In specific cases, (1) can be interpreted in a simple way. For example, when *A* is differentiable with derivative , so that , then the dual projections are given by and . More generally, if *A* is right-continuous with finite variation, then the infinitesimal increments can be interpreted in terms of Lebesgue-Stieltjes integrals. However, as the optional and predictable projections are defined for real valued processes, and is viewed as a stochastic measure, the right-hand-side of (1) is still problematic. This can be rectified by multiplying by an arbitrary process , and making use of the transitivity property . Integrating over time gives the more meaningful expressions

In contrast to (1), these equalities can be used to give mathematically rigorous definitions of the dual projections. As usual, we work with respect to a complete filtered probability space , and processes are identified whenever they are equal up to evanescence. The terminology `*raw IV process*‘ will be used to refer to any right-continuous integrable process whose variation on the whole of has finite expectation. The use of the word `raw’ here is just to signify that we are not requiring the process to be adapted. Next, to simplify the expressions, I will use the notation for the integral of a process with respect to another process *A*,

Note that, whereas the integral is implicitly taken over the range and does not involve the time-zero value of , I have included the time-zero values of the processes in the definition of . This is not essential, and could be excluded, so long as we were to restrict to processes starting from zero. The existence and uniqueness (up to evanescence) of the dual projections is given by the following result.

Theorem 1 (Dual Projections)LetAbe a raw IV process. Then,

- There exists a unique raw IV process satisfying

(2) for all bounded measurable processes . We refer to as the

dual optional projectionofA.- There exists a unique raw IV process satisfying

(3) for all bounded measurable processes . We refer to as the

dual predictable projectionofA.

Furthermore, ifAis nonnegative and increasing then so are and .

The proof will be given further down in this post. For now, note that nowhere in the statement of theorem 1 did we require the dual optional projection to be an optional process or the dual predictable projection to be predictable. In fact, these properties are automatically satisfied.

Theorem 2LetAbe a raw IV process. Then,

- is the unique optional IV process satisfying

(4) for all bounded optional processes .

- is the unique predictable IV process satisfying

(5) for all bounded predictable processes .

This can be used as an alternative definition of the dual projections, rather than theorem 1, which may be preferred as it explicitly makes clear that the dual projections are, respectively, optional and predictable. The proof of theorem 2 is a bit tricky, so will again be left until further down in this post. For now, we will show how localization can be used to relax the rather stringent condition that *A* has integrable variation.

Recall that for a cadlag process *A* and stopping time , the *stopped* process is defined by , and the *pre-stopped* process is defined by for and for . We say that *A* is *locally (raw) IV* if there exists a sequence of stopping times increasing to infinity such that are raw IV processes, and is *prelocally (raw) IV* if there exists a sequence of stopping times tending to infinity such that are raw IV processs. It should be clear that every raw IV process is locally IV, and that every locally IV process is also prelocally IV.

The dual optional projection can be extended to prelocally IV processes using a similar definition to that given for IV processes by Theorems 1 and 2. The difference is that we do not know, a-priori, that integrals of the form are well-defined and integral. Instead, we must impose this as a condition. In the following, when we state that is IV then we are saying both that is *A* integrable and that is a raw IV process. That is,

Also, as used elsewhere in these notes, a process will be said to be FV if it is cadlag, adapted, and of finite variation on all finite time intervals so that, in particular, adapted locally IV and prelocally IV processes are FV. When we say that a process has optional projection or predictable projection , we are implicitly including the condition that satisfies the necessary integrability properties for these projections to exist.

Theorem 3 (Dual Optional Projection)LetAbe prelocally a raw IV process. Then there exists a unique prelocally IV process satisfying

(6) for all processes with optional projection such that and are both IV. Equivalently, is the unique optional FV process such that, if is an optional process such that is IV, then is also IV and,

(7)

*Proof:* Letting *V* be the variation process of *A*,

(8) |

then, as *A* is prelocally a raw IV process, there exists a sequence of stopping times increasing to infinity such that are integrable. Choose a sequence of positive reals such that is finite. Then, we can construct a positive optional process

such that

is integrable. This implies that is an IV process and, by theorem 1, has a dual optional projection . Define

As is bounded by over the interval , this integral is well-defined and is an IV process, so is prelocally IV. Also, as is optional and is optional, hence adapted, it follows that is adapted or, equivalently, is optional. We show that satisfies (6) and (7).

Suppose that is a process with optional projection such that and are IV. We start by assuming that is bounded. Then, there exists a sequence of processes with and such that is bounded. For example, take . So and by dominated convergence for optional projections, . Therefore,

(9) |

Dominated convergence was applied in the first and fourth equalities, and the definition of the dual optional projection was used in the third. The condition that is bounded can be removed by, instead, choosing a sequence of optional processes such that and are bounded. For example, we can take . Then,

The first and third equalities here are applications of dominated convergence, and the second is using (9). This shows that (6) holds.

Next, suppose that is an optional process such that is IV. Using the fact that , (7) would follow immediately from (6) if it was known that is also IV. To show this, choose an optional such that is IV. For example, we can choose so that is bounded. Then, for any bounded process , applying (6) gives,

As this holds for all bounded processes ,

Then, as we can cancel it from both sides of the equality (i.e., integrate wrt both sides) to see that is -integrable and

is IV, as required.

Only the two uniqueness statements of the theorem remain to be established. Suppose that *B* is prelocally an IV process satisfying for all processes such that and are IV. As in the construction of above, we can find an optional process such that is IV. Replacing by if necessary, we can assume that is also IV. Then, for bounded processes ,

so, by Theorem 1, , and is uniquely determined.

Finally, suppose that *B* is an optional FV process such that is IV and for all optional processes such that is IV. In particular, with as above, and are IV. Furthermore, as *B* is adapted and is optional, it follows that is adapted and, hence, optional. For bounded optional ,

Theorem 2 then says that and, hence, is uniquely determined. ⬜

In a similar way, the dual predictable projection can be defined for all locally IV processes.

Theorem 4 (Dual Predictable Projection)LetAbe locally a raw IV process. Then there exists a unique locally IV process satisfying

(10) for all processes with predictable projection such that and are both IV. Equivalently, is the unique predictable FV process such that, if is a predictable process such that is IV, then is also IV and,

(11)

*Proof:* Letting *V* be the variation of *A*, given by (8), the condition that *A* is locally IV means that there is a sequence of stopping times increasing to infinity such that are integrable. As in the proof of theorem 3, we choose a sequence of positive reals such that is finite, and define the predictable process

The remainder of the proof follows exactly as for theorem 3, replacing `optional’ by `predictable’, `prelocally IV’ by `locally IV’, and the indicator processes by . One point to mention is that the proof of theorem 3 made use of the simple fact that an integral is optional (i.e., adapted) whenever and *A* are optional. Then, the current proof, requires the fact that is predictable whenever and *A* are predictable, which is only slightly less obvious. In fact, this statement reduces to showing that is adapted and that is predictable, which follows from the fact that it is the product of predictable processes. ⬜

#### -measures

It is possible to construct the dual projections satisfying the conclusions of theorem 1 is a relatively direct fashion. For a bounded random variable *U*, let *M* be the martingale given by and with right-continuous paths (at least, outside of a countable subset of ). The process with constant paths equal to *U* has optional projection *M* and predictable projection . From this, we obtain

This is sufficient to uniquely determine both and up to almost-sure equivalent and, by right-continuity, determines and up to evanescence. Furthermore, and can be constructed from the expressions above for and by taking Radon-Nikodym derivatives. However, I will use a more structured approach applying -measures, although the underlying ideas are the same as just described. This has some benefits as -measures are also useful in related areas such as with Doléans measures and the Doob-Meyer decomposition.

Use to denote the product sigma-algebra on . In relation to the optional and predictable sigma-algebras, we have the inclusions

Definition 5A -measure is a finite signed measure on (respectively, , ) which vanishes on evanescent sets. Furthermore, a -measure on is called

optionalif for all bounded processes .predictableif for all bounded processes .

Note that the requirement for to vanish on evanescent sets is necessary for expressions such as and to make sense, as the projections are only defined up to evanescence. The following is the main result on -measures, and relates them to IV processes. Unless explicitly stated otherwise, any -measures will be assumed to be defined on .

Theorem 6A raw IV process,A, uniquely defines a -measure, , given by

(12) for all bounded measurable processes . Furthermore, is a positive measure if and only if

Ais nonnegative and increasing.

Conversely, for any -measure , there exists a raw IV processA, uniquely defined up to evanescence, such that .

*Proof:* If *A* is a raw IV process then, to show that defined by (12) is a measure, we just need to verify that it is linear and satisfies monotone convergence. However, these follow immediately from linearity and monotone convergence for the integral with respect to *A* and for the expectation. Next, if is evanescent, then it is zero outside of a set of zero probability. So, is zero outside of *S*, and its expectation is zero. So, vanishes on evanescent processes as required. Furthermore, for any time and bounded random variable *U*,

which defines uniquely up to almost-sure equivalence. As *A* is right-continuous, this uniquely specifies *A* up to evanescence. It is clear that is a positive measure whenever *A* is nonnegative and increasing.

Conversely, suppose that is a nonnegative -measure and, for each , define a measure on by

for bounded random variables *U*. This is clearly nonnegative for , is linear, and satisfies monotone convergence. So is indeed a positive measure. Also, as is a -measure, we have whenever almost surely. So, is absolutely continuous with respect to . We can define a process *A* by the Radon-Nikodym derivative

so that is a nonnegative integrable random variable. For any nonnegative *U* and times , we have

So, almost surely. By countable additivity of probability measures, this means that is almost-surely increasing as *t* runs over the nonnegative rationals. Define the nonnegative, right-continuous and increasing process

As is the infimum of a countable set of random variables, it is measurable. Also, as *A* is almost surely increasing on the rationals, for any fixed we can choose a sequence of rationals tending to *t*, and, almost surely. Then, for any bounded random variable *U*,

This shows that almost surely, showing that *A* has a right-continuous and increasing modification. We suppose that we have chosen such a modification — that is, . Next, for a bounded random variable *U* and , let be the measurable process ,

As this is bounded for all and for all , this shows that *A* is a nonnegative increasing raw IV process and

The functional monotone class theorem extends this to all bounded measurable , so that .

Next, suppose that is a -measure, not assumed to be positive. By the Jordan decomposition, we can write for positive -measures and . By the argument above, there are increasing raw IV processes and with and . So, is a raw IV process satisfying . ⬜

In the following, and are referred to, respectively, as the optional and predictable projections of .

Theorem 7Any -measure on extends uniquely to an optional -measure, , on . Furthermore, is the unique -measure satisfying

(13) for bounded measurable . Similarly, any -measure on extends uniquely to a predictable -measure, , on . This is the unique -measure satisfying

(14)

for bounded measurable .

*Proof:* We start with the first statement, where is a -measure on . Define by (13). Recall that the optional projection is uniquely defined up to evanescence. As -measures vanish on evanescent sets, this implies that does not depend on the particular modification of the projection used and, so, is uniquely defined by (13). It just needs to be shown that it is a -measure. If is evanescent then, the projection is also evanescent and, so, vanishes. It only remains to demonstrate that is a (signed) measure. For this we need to show that linearity and dominated convergence hold.

Linearity follows immediately from linearity of the optional projection of . For monotone convergence, suppose that is a sequence of bounded measurable processes increasing to a bounded limit . By dominated convergence for the projection, increases to . By dominated convergence for the measure ,

as required.

Finally, it needs to be shown that is the unique extension of to an optional -measure. Clearly, as for optional , we have on the optional sigma-algebra. So, is an extension of to a -measure on . From (13),

showing that is optional. Conversely, suppose that is an extension of to an optional -measure on . From the definitions,

verifying that (13) holds.

For the second statement, where is a -measure on , the same argument holds replacing `optional’ by `predictable’. ⬜

The existence of the optional and predictable projections of a -measure on is an immediate consequence of Theorem 7. Equations (15) below are just restatements of (13) and (14).

Definition 8Let be a -measure (on ). We define its optional projection to be the unique optional -measure agreeing with on , and its predictable projection, , to be the unique predictable -measure agreeing with on .Equivalently, and are the unique -measures satisfying

(15)

for all bounded measurable .

Using Theorem 6 to translate the definition of the optional and predictable projections of -measures into corresponding projections of raw IV processes immediately gives Theorem 1.

*Proof of Theorem **1:* Let be the measure defined by theorem 6, and , be its optional and predictable projections, which are well-defined by theorem 7. Again, applying theorem 6, there exist unique IV processes and such that

Identities (2) and (3) are equivalent to (15) applied to ,

for all bounded measurable processes . Furthermore, if *A* is nonnegative and increasing then and are positive measures, so and are nonnegative and increasing. ⬜

We still need to prove theorem 2 and, in particular, show that dual optional projections are optional and that dual predictable projections are predictable. Starting with the dual optional projection, we prove the following result which relates optionality of a process with optionality of its associated -measure. This is simplified a bit by the observation that, for right-continuous processes, being optional is equivalent to being adapted.

Theorem 9LetAbe a raw IV process. Then, the following are equivalent,

- is optional.
- is adapted.
- For any bounded measurable random variable
Uand ,where

Mis the martingale with right-continuous paths (outside of a countable subset of ).- for all bounded measurable .
- is an optional -measure.

*Proof:*

*1 ⇔ 2:* We already know that optional processes are adapted and, conversely, that cadlag adapted processes are optional.

*2 ⇒ 4:* We use a little change of variables trick to prove this implication. Suppose, first, that *A* is nonnegative and increasing. If, for each , we let be the first time *t* at which ,

(16) |

This is increasing and is a stopping time as, for each , the event is equal to . By a change of variables,

for all bounded measurable . Here, we are adopting the convention that is equal to zero at time . From the definition of the optional projection, is equal to and,

as required. Now suppose that *A* is any adapted IV process. The idea is that the variation *V* of *A* can be expressed as the limit as *n* goes to infinity of the approximations,

(17) |

Here, is an increasing sequence of times with mesh going to zero as *n* goes to infinity. As the are clearly adapted, *V* is adapted. Decomposing for nonnegative adapted and increasing IV processes , and applying the argument above to , proves the implication for *A*.

*4 ⇒ 3:* Using the fact that *M* is the optional projection of the constant process equal to *U* at all times,

as required.

*3 ⇒ 2:* Letting , the martingale *M* defined by also satisfies at times . So,

from which it follows that *A* is adapted.

*4 ⇔ 5:* This is immediate from the definition (12) of . ⬜

We can similarly relate predictability of a process to predictability of its associated -measure. This is a bit more difficult than for optionality, as we do not have the simple equivalence of predictability of a process with being adapted. Instead of being adapted, we have the second statement of the theorem below. Recall that we have seen this same condition for a cadlag process to be predictable before in these notes although, there, some use was made of stochastic integration with respect to martingale integrators. In keeping with the current topic, I give an independent proof here which does not make use of advanced methods of stochastic calculus other than predictable projection.

Theorem 10LetAbe a raw IV process. Then, the following are equivalent,

- is predictable.
- is -measurable for every predictable stopping time , and (a.s.) for every totally inaccessible stopping time .
- For any bounded measurable random variable
Uand ,where

Mis the martingale with right-continuous paths (outside of a countable subset of ).- for all bounded measurable .
- is a predictable -measure.

*Proof:*

*1 ⇒ 4:* We use the same change of variables trick as in the proof of theorem 9. Supposing that *A* is increasing then, for each , we let be the first time *t* at which , as defined by (16). As *A* is increasing, the stochastic interval is equal to and, so, is predictable. Hence, is a predictable stopping time. From the definition of predictable projection, is equal to and, using a change of variables,

as required. Now suppose that *A* is any predictable IV process. It is trivial that *A* stopped at a fixed time *s* is predictable. Then, expression (17) for the approximations to the variation *V* of *A* gives predictable processes and, hence, *V* is predictable. Decomposing for nonnegative predictable and increasing IV processes , and applying the argument above to , proves the implication for *A*.

*4 ⇒ 3:* Using the fact that is the predictable projection of the constant process equal to *U* at all times,

as required.

*3 ⇒ 4:* If is of the form for a time and bounded random variable *U*, then, defined as above is the predictable projection of the constant process equal to *U* at all times. So,

This extends to all bounded measurable by the functional monotone class theorem.

*4 ⇒ 2:* First, the argument used to show that *A* is adapted in theorem 10 also applies here, so *A* is adapted. In particular, for a predictable stopping time , this implies that is -measurable. Then, for a bounded random variable *U*, the predictable projection of is equal to . So,

showing that is -measurable and, hence, is -measurable.

Now let be a totally inaccessible stopping time so, by definition, for all predictable stopping times . Therefore, is almost surely zero at each predictable stopping time and, so, has predictable projection equal to zero. That is,

and, hence, almost surely.

*2 ⇒ 1:* As is -measurable at each fixed time *t*, *A* is adapted. Then, is a thin process and, hence, for a sequence of stopping times satisfying the properties that each is either predictable or totally inaccessible and that whenever . Then, for any *n* where is totally inaccessible, by assumption we have almost surely. So, without loss of generality, we can suppose that is predictable. As *A* is adapted, is -measurable. By assumption, is also -measurable. This implies that the process is predictable. Furthermore, being adapted and left-continuous, is predictable. Hence,

decomposes *A* as the sum of predictable processes, showing that *A* is predictable.

*4 ⇔ 5:* This is immediate from the definition (12) of . ⬜

Theorems 9 and 10 can be used to prove Theorem 2 showing, in particular, that optional projections are optional and predictable projections are predictable.

*Proof of Theorem **2:* For a raw IV process *A*, by the definition given in theorem 1, the dual optional projection satisfies,

for all bounded measurable processes . The equivalence of the first and fourth statements of Theorem 9 implies that is optional and, for any bounded optional ,

showing that (4) holds. Conversely, suppose that *B* is an optional IV process satisfying

for all bounded optional . The equivalence of the first and fourth statements of Theorem 9 gives

showing that .

Finally, the statements of the theorem regarding the dual predictable projection follow by the same argument as above, replacing `optional’ by `predictable’, by , by , and by invoking Theorem 10 instead of 9. ⬜

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