Almost Sure

8 August 16

Purely Discontinuous Local Martingales

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

\displaystyle  X = X^{\rm c} + X^{\rm d} (1)

where {X^{\rm c}} is a continuous local martingale with {X^{\rm c}_0=0} and {X^{\rm d}} is a purely discontinuous local martingale.

Proof: As the process {H=\Delta X} 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 {X^{\rm d}} with {\Delta X^{\rm d}=H=\Delta X}. We can take {X^{\rm d}_0=X_0} so that {X^{\rm c}=X-X^{\rm d}} is a continuous local martingale starting from 0.

If {X=\tilde X^{\rm c}+\tilde X^{\rm d}} is another such decomposition, then {\tilde X^{\rm d}} and {X^{\rm d}} have the same jumps and initial value so, by Lemma 3 of the previous post, {\tilde X^{\rm d}=X^{\rm d}}. ⬜

Throughout the remainder of this post, the notation {X^{\rm c}} and {X^{\rm d}} will be used to denote the continuous and purely discontinuous parts of a local martingale X, as given by decomposition (1). Using the notation {\mathcal{M}_{\rm loc}}, {\mathcal{M}_{{\rm loc},0}^{\rm c}} and {\mathcal{M}_{\rm loc}^{\rm d} } 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

\displaystyle  \mathcal{M}_{\rm loc} = \mathcal{M}_{{\rm loc},0}^{\rm c} \oplus \mathcal{M}_{\rm loc}^{\rm d}. (2)

That is, {\mathcal{M}_{\rm loc}} is the direct sum of {\mathcal{M}_{{\rm loc},0}^{\rm c}} and {\mathcal{M}_{\rm loc}^{\rm d}}. 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 {\mathcal{M}_{\rm loc}} into the direct sum of the closed subspace {\mathcal{M}_{{\rm loc},0}^{\rm c}} 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 {\Delta[X]=(\Delta X)^2} and {\Delta[X,Y]=\Delta X\Delta Y}, so that they can be decomposed into continuous and pure jump components,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle [X]_t &\displaystyle=[X]^c_t+\sum_{s\le t}(\Delta X_s)^2,\smallskip\\ \displaystyle [X,Y]_t &\displaystyle=[X,Y]^c_t+\sum_{s\le t}\Delta X_s\Delta Y_s. \end{array} (3)

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.

  1. X is purely discontinuous.
  2. {[X,Y]=0} for all continuous local martingales Y.
  3. {[X,Y]^c=0} for all local martingales Y.
  4. {[X]^c=0}.
  5. there exists a sequence {\{X^n\}_{n=1,2,\ldots}} of FV local martingales such that

    \displaystyle  {\mathbb E}\left[\sup_{t\ge0}(X^n_t-X_t)^2\right]\rightarrow0.

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25 July 16

Constructing Martingales with Prescribed Jumps

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 {\mathcal{M}_{\rm loc}^{\rm d}}. Clearly, any linear combination of purely discontinuous local martingales is purely discontinuous. I will investigate {\mathcal{M}_{\rm loc}^{\rm d}} 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,

\displaystyle  [X,M]_t=\sum_{s\le t}\Delta X_s\Delta M_s=0.

The first equality follows because X is an FV process, and the second because M is continuous. So, {XM=XM-[X,M]} is a local martingale and X is purely discontinuous. ⬜

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 {X_0=Y_0} and {\Delta X = \Delta Y}, then {X=Y}.

Proof: Setting {M=X-Y} we have {M_0=0} and {\Delta M = 0}. So, M is a continuous local martingale and {M^2= MX-MY} is a local martingale starting from zero. Hence, it is a supermartingale and we have

\displaystyle  {\mathbb E}[M_t^2]\le{\mathbb E}[M_0^2]=0.

So {M_t=0} almost surely and, by right-continuity, {M=0} up to evanescence. ⬜

Note that if X is a continuous local martingale, then the constant process {Y_t=X_0} has the same initial value and jumps as X. So Lemma 3 has the immediate corollary.

Corollary 4 Any local martingale which is both continuous and purely discontinuous is almost surely constant.

Recalling that the jump process, {\Delta X}, 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, {H=\Delta X} for a local martingale X if and only if

  1. {\sqrt{\sum_{s\le t}H_s^2}} is locally integrable.
  2. {{\mathbb E}[1_{\{\tau < \infty\}}H_\tau\;\vert\mathcal{F}_{\tau-}]=0} (a.s.) for all predictable stopping times {\tau}.

Furthermore, X can be chosen to be purely discontinuous with {X_0=0}, in which case it is unique.

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