The concept of a stopping times was introduced a couple of posts back. Roughly speaking, these are times for which it is possible to observe when they occur. Often, however, it is useful to distinguish between different types of stopping times. A random time for which it is possible to predict when it is about to occur is called a predictable stopping time. As always, we work with respect to a filtered probability space .
Definition 1 A map is a predictable stopping time if there exists a sequence of stopping times satisfying whenever .
Predictable stopping times are alternatively referred to as previsible. The sequence of times in this definition are said to announce . Note that, in this definition, the random time was not explicitly required to be a stopping time. However, this is automatically the case, as the following equation shows.
One way in which predictable stopping times occur is as hitting times of a continuous adapted process. It is easy to predict when such a process is about to hit any level, because it must continuously approach that value.
Theorem 2 Let be a continuous adapted process and be a real number. Then
is a predictable stopping time.
Proof: Let be the first time at which which, by the debut theorem, is a stopping time. This gives an increasing sequence bounded above by . Also, whenever and, by leftcontinuity, setting gives whenever . So, , showing that the sequence increases to . If then, by continuity, . So, whenever and the sequence announces . ⬜
In fact, predictable stopping times are always hitting times of continuous processes, as stated by the following result. Furthermore, by the second condition below, it is enough to prove the much weaker condition that a random time can be announced `in probability’ to conclude that it is a predictable stopping time.
Lemma 3 Suppose that the filtration is complete and is a random time. The following are equivalent.
 is a predictable stopping time.
 For any there is a stopping time satisfying

(1) 
 for some continuous adapted process .
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