As I have mentioned before in these notes, when working with processes in continuous time, it is important to select a good modification. Typically, this means that we work with processes which are left or right continuous. However, in general, it can be difficult to show that the paths of a process satisfy such pathwise regularity. In this post I show that for optional and predictable processes, the section theorems introduced in the previous post can be used to considerably simplify the situation. Although they are interesting results in their own right, the main application in these notes will be to optional and predictable projection. Once the projections are defined, the results from this post will imply that they preserve certain continuity properties of the process paths.
Suppose, for example, that we have a continuous-time process X which we want to show to be right-continuous. It is certainly necessary that, for any sequence of times decreasing to a limit , almost-surely tends to . However, even if we can prove this for every possible decreasing sequence , it does not follow that X is right-continuous. As a counterexample, if is any continuously distributed random time, then the process is not right-continuous. However, so long as the distribution of has no atoms, X is almost-surely continuous at each fixed time t. It is remarkable, then, that if we generalise to look at sequences of stopping times, then convergence in probability along decreasing sequences of stopping times is enough to guarantee everywhere right-continuity of the process. At least, it is enough so long as we restrict consideration to optional processes.
As usual, we work with respect to a complete filtered probability space . Two processes are considered to be the same if they are equal up to evanescence, and any pathwise property is said to hold if it holds up to evanescence. That is, a process is right-continuous if and only is it is everywhere right-continuous on a set of probability 1. All processes will be taken to be real-valued, and a process is said to have left (or right) limits if its left (or right) limits exist everywhere, up to evanescence, and are finite.
Theorem 1 Let X be an optional process. Then,
- X is right-continuous if and only if in probability, for each uniformly bounded sequence of stopping times decreasing to a limit .
- X has right limits if and only if converges in probability, for each uniformly bounded decreasing sequence of stopping times.
- X has left limits if and only if converges in probability, for each uniformly bounded increasing sequence of stopping times.
The `only if’ parts of these statements is immediate, since convergence everywhere trivially implies convergence in probability. The importance of this theorem is in the `if’ directions. That is, it gives sufficient conditions to guarantee that the sample paths satisfy the respective regularity properties.
Note that conditions for left-continuity are absent from the statements of Theorem 1. In fact, left-continuity does not follow from the corresponding property along sequences of stopping times. Consider, for example, a Poisson process, X. This is right-continuous but not left-continuous. However, its jumps occur at totally inaccessible times. This implies that, for any sequence of stopping times increasing to a finite limit , it is true that converges almost surely to . In light of such examples, it is even more remarkable that right-continuity and the existence of left and right limits can be determined by just looking at convergence in probability along monotonic sequences of stopping times. Theorem 1 will be proven below, using the optional section theorem.
Theorem 2 Let X be a predictable process. Then,
- X is right-continuous if and only if in probability, for each uniformly bounded sequence of predictable stopping times decreasing to a limit .
- X is left-continuous if and only if in probability, for each uniformly bounded sequence of predictable stopping times increasing to a limit .
- X has right limits if and only if converges in probability, for each uniformly bounded decreasing sequence of predictable stopping times.
- X has left limits if and only if converges in probability, for each uniformly bounded increasing sequence of predictable stopping times.