In the previous post I introduced the definitions of the dual optional and predictable projections, firstly for processes of integrable variation and, then, generalised to processes which are only required to be locally (or prelocally) of integrable variation. We did not look at the properties of these dual projections beyond the fact that they exist and are uniquely defined, which are significant and important statements in their own right.
To recap, recall that an IV process, A, is right-continuous and such that its variation
is integrable at time , so that . The dual optional projection is defined for processes which are prelocally IV. That is, A has a dual optional projection if it is right-continuous and its variation process is prelocally integrable, so that there exist a sequence of stopping times increasing to infinity with integrable. More generally, A is a raw FV process if it is right-continuous with almost-surely finite variation over finite time intervals, so (a.s.) for all . Then, if a jointly measurable process is A-integrable on finite time intervals, we use
to denote the integral of with respect to A over the interval , which takes into account the value of at time 0 (unlike the integral which, implicitly, is defined on the interval ). In what follows, whenever we state that has any properties, such as being IV or prelocally IV, we are also including the statement that is A-integrable so that is a well-defined process. Also, whenever we state that a process has a dual optional projection, then we are also implicitly stating that it is prelocally IV.
From theorem 3 of the previous post, the dual optional projection is the unique prelocally IV process satisfying
for all optional such that is IV, in which case is also IV so that the expectations in this identity are well-defined.
I now look at the elementary properties of dual optional projections, as well as the corresponding properties of dual predictable projections. The most important property is that, according to the definition just stated, the dual projection exists and is uniquely defined. By comparison, the properties considered in this post are elementary and relatively easy to prove. So, I will simply state a theorem consisting of a list of all the properties under consideration, and will then run through their proofs. Starting with the dual optional projection, the main properties are listed below as Theorem 1.
Note that the first three statements are saying that the dual projection is indeed a linear projection from the prelocally IV processes onto the linear subspace of optional FV processes. As explained in the previous post, by comparison with the discrete-time setting, the dual optional projection can be expressed, in a non-rigorous sense, as taking the optional projection of the infinitesimal increments,
As is interpreted via the Lebesgue-Stieltjes integral , it is a random measure rather than a real-valued process. So, the optional projection of appearing in (2) does not really make sense. However, Theorem 1 does allow us to make sense of (2) in certain restricted cases. For example, if A is differentiable so that for a process , then (9) below gives . This agrees with (2) so long as is interpreted to mean . Also, restricting to the jump component of the increments, , (2) reduces to (11) below.
We defined the dual projection via expectations of integrals with the restriction that this is IV. An alternative approach is to first define the dual projections for IV processes, as was done in theorems 1 and 2 of the previous post, and then extend to (pre)locally IV processes by localisation of the projection. That this is consistent with our definitions follows from the fact that (pre)localisation commutes with the dual projection, as stated in (10) below.
- A raw FV process A is optional if and only if exists and is equal to A.
- If the dual optional projection of A exists then,
- If the dual optional projections of A and B exist, and , are -measurable random variables then,
- If the dual optional projection exists then is almost-surely finite and
- If U is a random variable and is a stopping time, then is prelocally IV if and only if is almost surely finite, in which case
- If the prelocally IV process A is nonnegative and increasing then so is and,
for all nonnegative measurable with optional projection . If A is merely increasing then so is and (7) holds for nonnegative measurable with .
- If A has dual optional projection and is an optional process such that is prelocally IV then, is -integrable and,
- If A is an optional FV process and is a measurable process with optional projection such that is prelocally IV then, is A-integrable and,
- If A has dual optional projection and is a stopping time then,
- If the dual optional projection exists, then its jump process is the optional projection of the jump process of A,
- If A has dual optional projection then
for all nonnegative measurable with optional projection .
- Let be a sequence of right-continuous processes with variation
If is prelocally IV then,