The previous post introduced the idea of a purely discontinuous local martingale. In the context of that post, such processes were used to construct local martingales with prescribed jumps, and enabled us to obtain uniqueness in the constructions given there. However, purely discontinuous local martingales are a very useful concept more generally in martingale and semimartingale theory, so I will go into more detail about such processes now. To start, we restate the definition from the previous post.
Definition 1 A local martingale X is said to be purely discontinuous iff XM is a local martingale for all continuous local martingales M.
We can show that every local martingale decomposes uniquely into continuous and purely discontinuous parts. Continuous local martingales are well understood — for instance, they can always be realized as time-changed Brownian motions. On the other hand, as we will see in a moment, purely discontinuous local martingales can be realized as limits of FV processes, and arguments involving FV local martingales can often to be extended to the purely discontinuous case. So, decomposition (1) below is useful as it allows arguments involving continuous-time local martingales to be broken down into different approaches involving their continuous and purely discontinuous parts. As always, two processes are considered to be equal if they are equivalent up to evanescence.
Theorem 2 Every local martingale X decomposes uniquely as
where is a continuous local martingale with and is a purely discontinuous local martingale.
Proof: As the process is, by definition, equal to the jump process of a local martingale then it satisfies the hypothesis of Theorem 5 of the previous post. So, there exists a purely discontinuous local martingale with . We can take so that is a continuous local martingale starting from 0.
If is another such decomposition, then and have the same jumps and initial value so, by Lemma 3 of the previous post, . ⬜
Throughout the remainder of this post, the notation and will be used to denote the continuous and purely discontinuous parts of a local martingale X, as given by decomposition (1). Using the notation , and respectively for the spaces of local martingales, continuous local martingales starting from zero and the purely discontinuous local martingales, Theorem 2 can be expressed succinctly as
That is, is the direct sum of and . Definition 2 identifies the purely discontinuous local martingales to be, in a sense, orthogonal to the continuous local martingales. Then, (2) can be understood as the decomposition of into the direct sum of the closed subspace and its orthogonal complement. This does in fact give an alternative, elementary, and commonly used, method of proving decomposition (1). As we have already shown the rather strong result of Theorem 5 from the previous post, the quickest way of proving the decomposition was to simply apply this result. I’ll give more details on the more elementary approach further below.
Definition 1 used above for the class of purely discontinuous local martingales was very convenient for our purposes, as it leads immediately to the proof of Theorem 2. However, there are many alternative characterizations of such processes. For example, they are precisely the processes which are limits of FV local martingales in a strong enough sense. They can also be characterized in terms of their quadratic variations and covariations. Recall that the quadratic variation and covariation are FV processes with jumps and , so that they can be decomposed into continuous and pure jump components,
The following theorem gives several alternative characterizations of the class of purely discontinuous local martingales.
Theorem 3 For a local martingale X, the following are equivalent.
- X is purely discontinuous.
- for all continuous local martingales Y.
- for all local martingales Y.
- there exists a sequence of FV local martingales such that