In this post, I introduce the concept of optional and predictable projections of jointly measurable processes. Optional projections of right-continuous processes and predictable projections of left-continuous processes were constructed in earlier posts, with the respective continuity conditions used to define the projection. These are, however, just special cases of the general theory. For arbitrary measurable processes, the projections cannot be expected to satisfy any such pathwise regularity conditions. Instead, we use the measurability criteria that the projections should be, respectively, optional and predictable.

The projection theorems are a relatively straightforward consequence of optional and predictable section. However, due to the difficulty of proving the section theorems, optional and predictable projection is generally considered to be an advanced or hard part of stochastic calculus. Here, I will make use of the section theorems as stated in an earlier post, but leave the proof of those until after developing the theory of projection.

As usual, we work with respect to a complete filtered probability space , and only consider real-valued processes. Any two processes are considered to be the same if they are equal up to evanescence. The optional projection is then defined (up to evanescence) by the following.

Theorem 1 (Optional Projection)LetXbe a measurable process such that is almost surely finite for each stopping time . Then, there exists a unique optional process , referred to as the optional projection ofX, satisfying

(1) almost surely, for each stopping time .

Predictable projection is defined similarly.

Theorem 2 (Predictable Projection)LetXbe a measurable process such that is almost surely finite for each predictable stopping time . Then, there exists a unique predictable process , referred to as the predictable projection ofX, satisfying

(2) almost surely, for each predictable stopping time .