In this post I give a proof of the theorems of optional and predictable section. These are often considered among the more advanced results in stochastic calculus, and many texts on the subject skip their proofs entirely. The approach here makes use of the measurable section theorem but, other than that, is relatively self-contained and will not require any knowledge of advanced topics beyond basic properties of probability measures.
Given a probability space we denote the projection map from to by
For a set then, by construction, for every there exists a with . Measurable section states that this choice can be made in a measurable way. That is, assuming that the probability space is complete, is measurable and there is a measurable section satisfying . I use the shorthand to mean , and it is convenient to extend the domain of to all of by setting outside of . So, we consider random times taking values in the extended nonnegative real numbers . The property that whenever can be expressed by stating that the graph of is contained in S, where the graph is defined as
The optional section theorem is a significant extension of measurable section which is very important to the general theory of stochastic processes. It starts with the concept of stopping times and with the optional sigma-algebra on . Then, it says that if S is optional its section can be chosen to be a stopping time. However, there is a slight restriction. It might not be possible to define such everywhere on , but instead only up to a set of positive probability , where can be made as small as we like. There is also a corresponding predictable section theorem, which says that if S is in the predictable sigma-algebra, its section can be chosen to be a predictable stopping time.
I give precise statements and proofs of optional and predictable section further below, and also prove a much more general section theorem which applies to any collection of random times satisfying a small number of required properties. Optional and predictable section will follow as consequences of this generalised section theorem.
Both the optional and predictable sigma-algebras, as well as the sigma-algebra used in the generalised section theorem, can be generated by collections of stochastic intervals. Any pair of random times defines a stochastic interval,
The debut of a set is defined to be the random time
In general, even if S is measurable, its debut need not be, although it can be shown to be measurable in the case that the probability space is complete. For a random time and a measurable set , we use to denote the restriction of to A defined by
We start with the general situation of a collection of random times satisfying a few required properties and show that, for sufficiently simple subsets of , the section can be chosen to be almost surely equal to the debut. It is straightforward that the collection of all stopping times defined with respect to some filtration do indeed satisfy the required properties for , but I also give a proof of this further below. A nonempty collection of subsets of a set X is called an algebra, Boolean algebra or, alternatively, a ring, if it is closed under finite unions, finite intersections, and under taking the complement of sets . Recall, also, that represents the countable intersections of A, which is the collection of sets of the form for sequences in .
Lemma 1 Let be a probability space and be a collection of measurable times satisfying,
- the constant function is in .
- and are in , for all .
- for all sequences in .
Then, letting be the collection of finite unions of stochastic intervals over , we have the following,
- is an algebra on .
- for all , its debut satisfies
and there is a with and almost surely.