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, if

Ais nonnegative and increasing then so are and .