The previous post introduced the concept of the compensator of a process, which is known to exist for all locally integrable semimartingales. In this post, I’ll just look at the very special case of compensators of processes consisting of a single jump of unit size.
Definition 1 Let be a stopping time. The compensator of is defined to be the compensator of .
So, the compensator A of is the unique predictable FV process such that and is a local martingale. Compensators of stopping times are sufficiently special that we can give an accurate description of how they behave. For example, if is predictable, then its compensator is just . If, on the other hand, is totally inaccessible and almost surely finite then, as we will see below, its compensator, A, continuously increases to a value which has the exponential distribution.
However, compensators of stopping times are sufficiently general to be able to describe the compensator of any cadlag adapted process X with locally integrable variation. We can break X down into a continuous part plus a sum over its jumps,
Here, are disjoint stopping times such that the union of their graphs contains all the jump times of X. That they are disjoint just means that whenever , for any . As was shown in an earlier post, not only is such a sequence of the stopping times guaranteed to exist, but each of the times can be chosen to be either predictable or totally inaccessible. As the first term, , on the right hand side of (1) is a continuous FV process, it is by definition equal to its own compensator. So, the compensator of X is equal to plus the sum of the compensators of . The reduces compensators of locally integrable FV processes to those of processes consisting of a single jump at either a predictable or a totally inaccessible time. Continue reading “Compensators of Stopping Times”