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 1Let 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,

(1) |

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”