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

${\sqrt{\sum_{s\le t}H_s^2}}$ is locally integrable.

${{\mathbb E}[1_{\{\tau < \infty\}}H_\tau\;\vert\mathcal{F}_{\tau-}]=0}$ (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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The Doob-Meyer Decomposition for Quasimartingales

As previously discussed, for discrete-time processes the Doob decomposition is a simple, but very useful, technique which allows us to decompose any integrable process into the sum of a martingale and a predictable process. If ${\{X_n\}_{n=0,1,2,\ldots}}$ is an integrable discrete-time process adapted to a filtration ${\{\mathcal{F}_n\}_{n=0,1,2,\ldots}}$ , then the Doob decomposition expresses X as

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle X_n&\displaystyle=M_n+A_n,\smallskip\\ \displaystyle A_n&\displaystyle=\sum_{k=1}^n{\mathbb E}\left[X_k-X_{k-1}\;\vert\mathcal{F}_{k-1}\right]. \end{array}$

(1)

Then, M is then a martingale and A is an integrable process which is also predictable, in the sense that ${A_n}$ is ${\mathcal{F}_{n-1}}$ -measurable for each ${n > 0}$ . The expected value of the variation of A can be computed in terms of X,

$\displaystyle {\mathbb E}\left[\sum_{k=1}^n\lvert A_k-A_{k-1}\rvert\right] ={\mathbb E}\left[\sum_{k=1}^n\left\lvert {\mathbb E}[X_k-X_{k-1}\vert\;\mathcal{F}_{k-1}]\right\rvert\right].$

This is the mean variation of X.

In continuous time, the situation is rather more complex, and will require constraints on the process X other than just integrability. We have already discussed the case for submartingales — the Doob-Meyer decomposition. This decomposes a submartingale into a local martingale and a predictable increasing process.

A natural setting for further generalising the Doob-Meyer decomposition is that of quasimartingales. In continuous time, the appropriate class of processes to use for the component A of the decomposition is the predictable FV processes. Decomposition (2) below is the same as that in the previous post on special semimartingales. This is not surprising, as we have already seen that the class of special semimartingales is identical to the class of local quasimartingales. The difference with the current setting is that we can express the expected variation of A in terms of the mean variation of X, and obtain a necessary and sufficient condition for the local martingale component to be a proper martingale.

As was noted in an earlier post, historically, decomposition (2) for quasimartingales played an important part in the development of stochastic calculus and, in particular, in the proof of the Bichteler-Dellacherie theorem. That is not the case in these notes, however, as we have already proven the main results without requiring quasimartingales. As always, any two processes are identified whenever they are equivalent up to evanescence.

Theorem 1 Every cadlag quasimartingale X uniquely decomposes as

$\displaystyle X=M+A$ (2)

where M is a local martingale and A is a predictable FV process with ${A_0=0}$ . Then, A has integrable variation over each finite time interval ${[0,t]}$ satisfying

$\displaystyle {\rm Var}_t(X)={\rm Var}_t(M)+{\mathbb E}\left[\int_0^t\,\vert dA\vert\right].$ (3)

so that, in particular,

$\displaystyle {\mathbb E}\left[\int_0^t\,\vert dA\vert\right]\le{\rm Var}_t(X).$ (4)

Furthermore, the following are equivalent,

X is of class (DL).

M is a proper martingale.

inequality (4) is an equality for all times t.

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Properties of Quasimartingales

The previous two posts introduced the concept of quasimartingales, and noted that they can be considered as a generalization of submartingales and supermartingales. In this post we prove various basic properties of quasimartingales and of the mean variation, extending results of martingale theory to this situation.

We start with a version of optional stopping which applies for quasimartingales. For now, we just consider simple stopping times, which are stopping times taking values in a finite subset of the nonnegative extended reals ${\bar{\mathbb R}_+=[0,\infty]}$ . Stopping a process can only decrease its mean variation (recall the alternative definitions ${{\rm Var}}$ and ${{\rm Var}^*}$ for the mean variation). For example, a process X is a martingale if and only if ${{\rm Var}(X)=0}$ , so in this case the following result says that stopped martingales are martingales.

Lemma 1 Let X be an adapted process and ${\tau}$ be a simple stopping time. Then

$\displaystyle {\rm Var}^*(X^\tau)\le{\rm Var}^*(X).$ (1)

Assuming, furthermore, that X is integrable,

$\displaystyle {\rm Var}(X^\tau)\le{\rm Var}(X).$ (2)

and, more precisely,

$\displaystyle {\rm Var}(X)={\rm Var}(X^\tau)+{\rm Var}(X-X^\tau)$ (3)

Continue reading “Properties of Quasimartingales” →

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