In this post we will describe precisely which processes can be realized as the jumps of a local martingale. This leads to very useful decomposition results for processes — see Theorem 10 below, where we give a decomposition of a process X into martingale and predictable components. As I will explore further in future posts, this enables us to construct particularly useful decompositions for local martingales and semimartingales.
Before going any further, we start by defining the class of local martingales which will be used to match prescribed jump processes. The purely discontinuous local martingales are, in a sense, the orthogonal complement to the class of continuous local martingales.
Definition 1 A local martingale X is said to be purely discontinuous iff XM is a local martingale for all continuous local martingales M.
The class of purely discontinuous local martingales is often denoted as . Clearly, any linear combination of purely discontinuous local martingales is purely discontinuous. I will investigate in more detail later but, in order that we do have plenty of examples of such processes, we show that all FV local martingales are purely discontinuous.
Lemma 2 Every FV local martingale is purely discontinuous.
Proof: If X is an FV local martingale and M is a continuous local martingale then we can compute the quadratic covariation,
Next, an important property of purely discontinuous local martingales is that they are determined uniquely by their jumps. Throughout these notes, I am considering two processes to be equal whenever they are equal up to evanescence.
Lemma 3 Purely discontinuous local martingales are uniquely determined by their initial value and jumps. That is, if X and Y are purely discontinuous local martingales with and , then .
Proof: Setting we have and . So, M is a continuous local martingale and is a local martingale starting from zero. Hence, it is a supermartingale and we have
So almost surely and, by right-continuity, up to evanescence. ⬜
Note that if X is a continuous local martingale, then the constant process has the same initial value and jumps as X. So Lemma 3 has the immediate corollary.
Corollary 4 Any local martingale with is both continuous and purely discontinuous is almost surely constant.
Recalling that the jump process, , of a cadlag adapted process X is thin, we now state the main theorem of this post and describe precisely those processes which occur as the jumps of a local martingale.
Theorem 5 Let H be a thin process. Then, for a local martingale X if and only if
Furthermore, X can be chosen to be purely discontinuous with , in which case it is unique.