The Doob-Meyer Decomposition
杜布-迈耶分解

Stochastic Calculus Notes, The General Theory of Semimartingales
乔治·洛瑟随机微积分笔记，半鞅的一般理论，2011 年 12 月 30 日

The Doob-Meyer decomposition was a very important result, historically, in the development of stochastic calculus. This theorem states that every cadlag submartingale uniquely decomposes as the sum of a local martingale and an increasing predictable process. For one thing, if X is a square-integrable martingale then Jensen’s inequality implies that {X^2} is a submartingale, so the Doob-Meyer decomposition guarantees the existence of an increasing predictable process {\langle X\rangle} such that {X^2-\langle X\rangle} is a local martingale. The term {\langle X\rangle} is called the predictable quadratic variation of X and, by using a version of the Ito isometry, can be used to define stochastic integration with respect to square-integrable martingales. For another, semimartingales were historically defined as sums of local martingales and finite variation processes, so the Doob-Meyer decomposition ensures that all local submartingales are also semimartingales. Going further, the Doob-Meyer decomposition is used as an important ingredient in many proofs of the Bichteler-Dellacherie theorem.
杜布-迈耶分解在历史上是随机微积分发展中一个非常重要的成果。该定理指出，每一个右连左极下鞅都可以唯一地分解为一个局部鞅和一个可预测递增过程之和。一方面，如果 X 是一个平方可积鞅，那么詹森不等式意味着 {X^2} 是一个下鞅，因此杜布-迈耶分解保证了存在一个可预测递增过程 {\langle X\rangle} ，使得 {X^2-\langle X\rangle} 是一个局部鞅。术语 {\langle X\rangle} 被称为 X 的可预测二次变差，并且通过使用伊藤等距的一个版本，可以用来定义关于平方可积鞅的随机积分。另一方面，半鞅在历史上被定义为局部鞅和有限变差过程的和，因此杜布-迈耶分解确保了所有局部下鞅也是半鞅。更进一步，杜布-迈耶分解被用作比希特勒-德拉谢里定理许多证明中的重要组成部分。

The approach taken in these notes is somewhat different from the historical development, however. We introduced stochastic integration and semimartingales early on, without requiring much prior knowledge of the general theory of stochastic processes. We have also developed the theory of semimartingales, such as proving the Bichteler-Dellacherie theorem, using a stochastic integration based method. So, the Doob-Meyer decomposition does not play such a pivotal role in these notes as in some other approaches to stochastic calculus. In fact, the special semimartingale decomposition already states a form of the Doob-Meyer decomposition in a more general setting. So, the main part of the proof given in this post will be to show that all local submartingales are semimartingales, allowing the decomposition for special semimartingales to be applied.
然而，这些笔记所采用的方法与历史发展有所不同。我们早期引入了随机积分和半鞅，而不需要太多关于随机过程一般理论的先验知识。我们还发展了半鞅理论，例如使用基于随机积分的方法证明了 Bichteler-Dellacherie 定理。因此，在这些笔记中，Doob-Meyer 分解并不像在其他随机微积分方法中那样起着关键作用。事实上，特殊半鞅分解已经在更一般的设置中陈述了 Doob-Meyer 分解的一种形式。因此，本文给出的证明的主要部分将是展示所有局部下鞅都是半鞅，从而允许应用特殊半鞅的分解。

The Doob-Meyer decomposition is especially easy to understand in discrete time, where it reduces to the much simpler Doob decomposition. 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
杜布-迈耶分解在离散时间中特别容易理解，它简化为更简单的杜布分解。如果 {\{X_n\}_{n=0,1,2,\ldots}} 是一个适应于过滤 {\{\mathcal{F}_n\}_{n=0,1,2,\ldots}} 的可积离散时间过程，那么杜布分解将 X 表示为

\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)

As previously discussed, 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}. Furthermore, X is a submartingale if and only if {{\mathbb E}[X_n-X_{n-1}\vert\mathcal{F}_{n-1}]\ge0} or, equivalently, if A is almost surely increasing.
如前所述，M 是一个鞅，A 是一个可积过程，并且也是可预测的，即对于每个 {n > 0} ， {A_n} 是 {\mathcal{F}_{n-1}} 可测的。此外，X 是一个下鞅当且仅当 {{\mathbb E}[X_n-X_{n-1}\vert\mathcal{F}_{n-1}]\ge0} ，或者等价地，如果 A 几乎肯定是递增的。

Moving to continuous time, we work with respect to a complete filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})} with time index t ranging over the nonnegative real numbers. Then, the continuous-time version of (1) takes A to be a right-continuous and increasing process which is predictable, in the sense that it is measurable with respect to the σ-algebra generated by the class of left-continuous and adapted processes. Often, the Doob-Meyer decomposition is stated under additional assumptions, such as X being of class (D) or satisfying some similar uniform integrability property. To be as general possible, the statement I give here only requires X to be a local submartingale, and furthermore states how the decomposition is affected by various stronger hypotheses that X may satisfy.
转向连续时间，我们在一个完备的过滤概率空间 {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})} 上工作，时间索引 t 覆盖非负实数。于是，(1)式的连续时间版本将 A 视为一个右连续且递增的过程，它是可预测的，即相对于由左连续且适应的过程类生成的σ-代数可测。通常，Doob-Meyer 分解是在额外假设下陈述的，例如 X 属于(D)类或满足某些类似的均匀可积性条件。为了尽可能一般化，我在此给出的陈述仅要求 X 是一个局部下鞅，并进一步说明了当 X 满足各种更强的假设时，分解会受到怎样的影响。

Theorem 1 (Doob-Meyer) Any local submartingale X has a unique decomposition
定理 1（Doob-Meyer）任何局部下鞅 X 都有唯一的分解

\displaystyle X=M+A, (2)

where M is a local martingale and A is a predictable increasing process starting from zero.
其中 M 是局部鞅，A 是从零开始的可预测递增过程。

Furthermore,   此外，

  1. if X is a proper submartingale, then A is integrable and satisfies
    如果 X 是一个真下鞅，则 A 是可积的且满足
    \displaystyle {\mathbb E}[A_\tau]\le{\mathbb E}[X_\tau-X_0] (3)

    for all uniformly bounded stopping times {\tau}.
    对于所有一致有界的停时 {\tau} 。

  2. X is of class (DL) if and only if M is a proper martingale and A is integrable, in which case
    X 属于 (DL) 类当且仅当 M 是一个真鞅且 A 是可积的，在这种情况下
    \displaystyle {\mathbb E}[A_\tau]={\mathbb E}[X_\tau-X_0] (4)

    for all uniformly bounded stopping times {\tau}.
    对于所有一致有界的停时 {\tau} 。

  3. X is of class (D) if and only if M is a uniformly integrable martingale and {A_\infty} is integrable. Then, {X_\infty=\lim_{t\rightarrow\infty}X_t} and {M_\infty=\lim_{t\rightarrow\infty}M_t} exist almost surely, and (4) holds for all (not necessarily finite) stopping times {\tau}.
    X 属于类(D)当且仅当 M 是一致可积鞅且 {A_\infty} 是可积的。那么， {X_\infty=\lim_{t\rightarrow\infty}X_t} 和 {M_\infty=\lim_{t\rightarrow\infty}M_t} 几乎必然存在，且(4)对所有（不一定有限）停时 {\tau} 成立。

Note that, by definition, local submartingales and local martingales are always cadlag processes and, hence, in decomposition (2), A is automatically required to be cadlag. So, right-continuity of A did not have to be explicitly stated. Also, Theorem 1 implies that any local submartingale X decomposes as a local martingale plus a finite variation process. As mentioned above, this implies that X is a semimartingale. Here, we work in the opposite direction, proving first that every local submartingale is a semimartingale and, then, use this to give a proof of Theorem 1. Recall that, in these notes, a cadlag adapted process X is defined to be a semimartingale if and only if the stochastic integral {\int\xi\,dX} is defined for all bounded predictable integrands {\xi}.
注意，根据定义，局部下鞅和局部鞅始终是右连左极过程（cadlag 过程），因此，在分解式（2）中，A 自动要求为 cadlag 过程。因此，A 的右连续性无需明确说明。此外，定理 1 暗示任何局部下鞅 X 可分解为一个局部鞅加上一个有限变差过程。如上所述，这意味着 X 是一个半鞅。在此，我们采取相反方向，首先证明每个局部下鞅都是半鞅，然后利用这一点来证明定理 1。回想一下，在这些笔记中，一个 cadlag 适应过程 X 被定义为半鞅，当且仅当对于所有有界可预测被积函数 {\xi} ，随机积分 {\int\xi\,dX} 被定义。

Lemma 2 Every local submartingale is a semimartingale.
引理 2 每一个局部下鞅都是半鞅。

Proof: It was prevously shown that X is a semimartingale if, for each fixed {t\in{\mathbb R}_+}, the set
证明：先前已证明，若对于每个固定的 {t\in{\mathbb R}_+} ，集合 X 是一个半鞅，则 X 是半鞅

\displaystyle \left\{\int_0^t\xi\,dX\colon\vert\xi\vert\le1,\;\xi{\rm\ is\ elementary}\right\} (5)

is bounded in probability. This characterization was also stated as part of the Bichteler-Dellacherie theorem, and is what we will use to show that a local submartingale X is a semimartingale. The proof is similar to the earlier proof that local martingales are semimartingales. By localization, we can suppose that X is a proper submartingale.
在概率上有界。这一特征也被表述为 Bichteler-Dellacherie 定理的一部分，并且我们将利用它来证明局部下鞅 X 是一个半鞅。该证明类似于之前证明局部鞅是半鞅的过程。通过局部化，我们可以假设 X 是一个适当的下鞅。

For any elementary process {\xi} and fixed time {t > 0}, there exist times {0=t_0 \le t_1\le\cdots \le t_n=t} such that
对于任何基本过程 {\xi} 和固定时间 {t > 0} ，存在时间 {0=t_0 \le t_1\le\cdots \le t_n=t} 使得

\displaystyle 1_{(0,t]}\xi=\sum_{k=1}^n\xi_{t_k}1_{(t_{k-1},t_k]},

and {\xi_{t_k}} are {\mathcal{F}_{t_{k-1}}}-measurable random variables. Consider the discrete-time process {\tilde X_k=X_{t_k}-X_0}, which is a submartingale adapted to the discrete filtration {\mathcal{\tilde F}_k=\mathcal{F}_{t_k}}. Applying the Doob decomposition (1) to {\tilde X}, we can write {\tilde X_k=M_k+A_k}, where M is a discrete-time martingale and A is increasing with {{\mathbb E}[A_n]={\mathbb E}[\tilde X_n]}. So, if {\vert\xi\vert\le1},
且 {\xi_{t_k}} 和 {\mathcal{F}_{t_{k-1}}} 是可测随机变量。考虑离散时间过程 {\tilde X_k=X_{t_k}-X_0} ，它是适应于离散过滤 {\mathcal{\tilde F}_k=\mathcal{F}_{t_k}} 的下鞅。将 Doob 分解(1)应用于 {\tilde X} ，我们可以写成 {\tilde X_k=M_k+A_k} ，其中 M 是离散时间鞅，A 随 {{\mathbb E}[A_n]={\mathbb E}[\tilde X_n]} 递增。因此，如果 {\vert\xi\vert\le1} ，

\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle{\mathbb P}\left(\left\vert\sum_{k=1}^n\xi_{t_k}(A_k-A_{k-1})\right\vert\ge K\right)&\displaystyle\le{\mathbb P}\left(A_n\ge K\right)\smallskip\\ &\displaystyle\le K^{-1}{\mathbb E}[A_n]\smallskip\\ &\displaystyle=K^{-1}{\mathbb E}[X_t-X_0] \end{array} (6)

for all {K > 0}. Also, as previously shown in the proof that martingales are semimartingales, there exists a constant {c > 0}, independent of choice of X, {\xi} and K, such that
对于所有 {K > 0} 。此外，正如先前在证明鞅是半鞅时所展示的，存在一个常数 {c > 0} ，与 X、 {\xi} 和 K 的选择无关，使得

\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle{\mathbb P}\left(\left\vert\sum_{k=1}^n\xi_{t_k}(M_k-M_{k-1})\right\vert\ge K\right)&\displaystyle\le\frac{c}{K}{\mathbb E}[\vert M_n\vert]\smallskip\\ &\displaystyle\le\frac{c}{K}{\mathbb E}\left[\vert \tilde X_n\vert + A_n\right]\smallskip\\ &\displaystyle\le\frac{2c}{K}{\mathbb E}\left[\vert X_t-X_0\vert\right]. \end{array} (7)

Combining (6) and (7) gives
结合（6）和（7）得出

\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rcl} \displaystyle{\mathbb P}\left(\left\vert\int_0^t\xi\,dX\right\vert > K\right)&\displaystyle\le&\displaystyle{\mathbb P}\left(\left\vert\sum_{k=1}^n\xi_{t_k}(A_k-A_{k-1})\right\vert\ge\frac{K}{2}\right)\smallskip\\ &&\displaystyle+{\mathbb P}\left(\left\vert\sum_{k=1}^n\xi_{t_k}(M_k-M_{k-1})\right\vert\ge\frac{K}{2}\right)\smallskip\\ &\displaystyle\le&\displaystyle\frac{4c+2}{K}{\mathbb E}\left[\vert X_t-X_0\vert\right]. \end{array}

So, choosing K large, the probability on the left hand side can be made as small as we like, independently of {\xi}. This shows that the set (5) is bounded in probability as required. ⬜
因此，选择足够大的 K，左侧的概率可以任意小，且与 {\xi} 无关。这表明集合(5)在概率上是有界的，符合要求。⬜

With Lemma 2 out of the way, all the ingredients are now in place to give a proof of the Doob-Meyer decomposition. This is just an application of the special semimartingale decomposition from an earlier post, although there is still a small amount of work required to show that the compensator A is indeed increasing. Let us start by proving the first part of Theorem 1 where it is only assumed that X is a local submartingale.
在解决了引理 2 之后，所有要素都已就绪，可以给出 Doob-Meyer 分解的证明了。这实际上是对之前帖子中特殊半鞅分解的应用，尽管仍需少量工作来证明补偿器 A 确实是递增的。让我们从证明定理 1 的第一部分开始，其中仅假设 X 是一个局部下鞅。

Lemma 3 Every local submartingale X uniquely decomposes as {X=M+A}, where M is a local martingale and A is an increasing predictable process starting from zero.
引理 3 每一个局部下鞅 X 唯一地分解为 {X=M+A} ，其中 M 是一个局部鞅，A 是一个从零开始的递增可预测过程。

Proof: By Lemma 2, we know that X is a semimartingale. Also, as it is a local submartingale, X is locally integrable. So, it uniquely decomposes as {X=M+A}, where M is a local martingale and A is a predictable FV process with {A_0=0}. It only remains to be shown that A is increasing.
证明：根据引理 2，我们知道 X 是一个半鞅。此外，由于它是一个局部下鞅，X 是局部可积的。因此，它可以唯一地分解为 {X=M+A} ，其中 M 是一个局部鞅，A 是一个可预测的有限变差过程，且满足 {A_0=0} 。剩下的只需证明 A 是递增的。

As {A=X-M} is a local submartingale, there exist stopping times {\tau_n} increasing to infinity such that {A^{\tau_n}} are submartingales. Also, as A is predictable, {\tau_n} can be chosen such that {A^{\tau_n}} has uniformly bounded and, hence, integrable total variation. Then, by the properties of submartingales,
由于 {A=X-M} 是局部下鞅，存在趋于无穷的停时 {\tau_n} 使得 {A^{\tau_n}} 为下鞅。此外，由于 A 是可预测的，可以选择 {\tau_n} 使得 {A^{\tau_n}} 具有一致有界且因此可积的总变差。然后，根据下鞅的性质，

\displaystyle {\mathbb E}\left[\int_0^{\tau_n}\vert\xi\vert\,dA\right]\ge0 (8)

for any bounded elementary process {\xi}. Let us denote the set of uniformly bounded predictable processes for which (8) holds by S. Then, as just shown, S contains all bounded elementary processes. Next, consider a sequence {\{\xi^m\}_{m=1,2,\ldots}} of predictable processes in S, uniformly bounded by a constant K, and converging to the limit {\xi}. As {A^{\tau_n}} has integrable total variation, dominated convergence gives
对于任何有界初等过程 {\xi} 。让我们用 S 表示满足(8)的一致有界可预测过程的集合。那么，正如刚才所示，S 包含所有有界初等过程。接下来，考虑 S 中一致被常数 K 界定的可预测过程序列 {\{\xi^m\}_{m=1,2,\ldots}} ，并收敛到极限 {\xi} 。由于 {A^{\tau_n}} 具有可积的总变差，由控制收敛定理可得

\displaystyle {\mathbb E}\left[\int_0^{\tau_n}\vert\xi\vert\,dA\right]=\lim_{m\rightarrow\infty}{\mathbb E}\left[\int_0^{\tau_n}\vert\xi^m\vert\,dA\right]\ge0.

So, {\xi\in S}. Therefore, by the monotone class theorem, S contains all uniformly bounded predictable processes.
因此，根据单调类定理，S 包含所有一致有界的可预测过程。

Now, as A is a predictable FV process, it can be written as {A=A^+-A^-} where {A^+} and {A^-} are increasing processes, and {A^-=-\int\xi\,dA} for some predictable process {0\le\xi\le1}. By (8),
现在，由于 A 是一个可预测的 FV 过程，它可以写成 {A=A^+-A^-} ，其中 {A^+} 和 {A^-} 是递增过程，且对于某个可预测过程 {0\le\xi\le1} ，有 {A^-=-\int\xi\,dA} 。根据(8)，

\displaystyle {\mathbb E}\left[A^-_{\tau_n}\right]=-{\mathbb E}\left[\int_0^{\tau_n}\xi\,dA\right]\le0.

Letting n increase to infinity, monotone convergence gives {{\mathbb E}[A^-_\infty]=0}. Finally, as it is nonnegative and increasing, we almost surely have {A^-_t=0} for all t and, hence, {A=A^+} is increasing. ⬜
令 n 趋于无穷大，单调收敛给出 {{\mathbb E}[A^-_\infty]=0} 。最后，由于它是非负且递增的，我们几乎肯定对于所有 t 都有 {A^-_t=0} ，因此 {A=A^+} 是递增的。⬜

To complete the proof of Theorem 1, just the `furthermore’ part remains to be shown. That is, under the additional hypotheses for X we obtain stronger properties satisfied by M and A. We prove this now. Note that nowhere in this part of the proof requires the fact that A is predictable, just that it is an increasing process starting from zero.
为了完成定理 1 的证明，仅剩“进一步”部分需要展示。即，在 X 的额外假设下，我们得到 M 和 A 满足的更强性质。我们现在证明这一点。注意，在证明的这一部分中，并不需要 A 是可预测的这一事实，仅需它是一个从零开始的递增过程。

Proof of Theorem 1:  定理 1 的证明：
Suppose that X is a proper submartingale. Then, choose stopping times {\tau_n} increasing to infinity such that the stopped processes {M^{\tau_n}} are proper martingales. For any uniformly bounded stopping time {\tau}, optional sampling says that {X_{\tau_n\wedge\tau}} is integrable with expectation bounded by {{\mathbb E}[X_\tau]}. So,
假设 X 是一个真下鞅。那么，选择趋于无穷的停时 {\tau_n} ，使得停止过程 {M^{\tau_n}} 成为真鞅。对于任何一致有界的停时 {\tau} ，可选抽样定理表明 {X_{\tau_n\wedge\tau}} 是可积的，其期望值以 {{\mathbb E}[X_\tau]} 为界。因此，

\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle{\mathbb E}[A_{\tau_n\wedge\tau}]&\displaystyle={\mathbb E}[X_{\tau_n\wedge\tau}-M^{\tau_n}_\tau]\smallskip\\ &\displaystyle\le{\mathbb E}[X_\tau-M_0]\smallskip\\ &\displaystyle={\mathbb E}[X_\tau-X_0]. \end{array}

Letting n increase to infinity and using monotone convergence gives inequality (3) so, in particular, A is integrable.
让 n 趋近于无穷大并利用单调收敛性，得到不等式（3），因此，特别地，A 是可积的。

Now, suppose that X is of class (DL). Then, it is a proper submartingale so, as shown above, A is integrable. As it is also nonnegative and increasing, A is dominated in {L^1} on each bounded interval. So, {M=X-A} is of class (DL) and, hence, is a proper martingale. Then, for any uniformly bounded stopping time {\tau}, optional sampling gives
现在，假设 X 属于(DL)类。那么，它是一个适当的亚鞅，因此如上所示，A 是可积的。由于它也是非负且递增的，A 在每个有界区间上在 {L^1} 中被支配。因此， {M=X-A} 属于(DL)类，因此是一个适当的鞅。然后，对于任何一致有界的停时 {\tau} ，可选抽样给出

\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle{\mathbb E}[A_\tau]&\displaystyle={\mathbb E}[X_\tau-M_\tau]\smallskip\\ &\displaystyle={\mathbb E}[X_\tau-M_0]\smallskip\\ &\displaystyle={\mathbb E}[X_\tau-X_0]. \end{array}

This proves equality (4).
这证明了等式（4）。

Conversely, suppose that M is a proper martingale and A is integrable. Then, M is of class (DL) and, as it is dominated in {L^1} on finite intervals, A is also of class (DL). Therefore, {X=M+A} is of class (DL).
反之，假设 M 是一个真鞅且 A 是可积的。那么，M 属于(DL)类，并且由于它在有限区间上被 {L^1} 所控制，A 也属于(DL)类。因此， {X=M+A} 属于(DL)类。

Now, suppose that X is of class (D). In particular, it is of class (DL) and, as shown above, M is a proper martingale and A is integrable. By submartingale convergence, the limit {X_\infty=\lim_{t\rightarrow\infty}X_t} exists almost-surely. Applying (3) and monotone convergence,
现在，假设 X 属于(D)类。特别地，它属于(DL)类，并且如上所示，M 是一个适当的鞅，A 是可积的。根据下鞅收敛性，极限 {X_\infty=\lim_{t\rightarrow\infty}X_t} 几乎必然存在。应用(3)和单调收敛性，

\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle{\mathbb E}[A_\tau]&\displaystyle=\lim_{t\rightarrow\infty}{\mathbb E}[A_{\tau\wedge t}]\smallskip\\ &\displaystyle=\lim_{t\rightarrow\infty}{\mathbb E}[X_{\tau\wedge t}-X_0]\smallskip\\ &\displaystyle={\mathbb E}[X_\tau-X_0]. \end{array}

The last equality here uses uniform integrability of {X_{\tau\wedge t}} over {t\ge0}, since X is of class (D). So, (3) holds for all stopping times and, by taking {\tau=\infty}, we see that {A_\infty} is integrable. As it is dominated in {L^1}, A is in class (D). So, {M=X-A} is also in class (D) and, hence, is a uniformly integrable martingale.
最后一个等式这里使用了 {X_{\tau\wedge t}} 在 {t\ge0} 上的一致可积性，因为 X 属于(D)类。因此，(3)对所有停时成立，通过取 {\tau=\infty} ，我们看到 {A_\infty} 是可积的。由于它在 {L^1} 中被支配，A 属于(D)类。因此， {M=X-A} 也属于(D)类，因此是一个一致可积的鞅。

Conversely, suppose that M is a uniformly integrable martingale and that {A_\infty} is integrable. Then, M is of class (D) and, as it is dominated in {L^1}, A is also of class (D). So, {X=M+A} is of class (D) as required. ⬜
反之，假设 M 是一致可积鞅且 {A_\infty} 可积。那么，M 属于(D)类，并且由于它在 {L^1} 中被控制，A 也属于(D)类。因此， {X=M+A} 按要求属于(D)类。⬜

Approximating the Compensator
近似补偿器

The process A appearing in decomposition (2) is called the compensator of X. By definition, it is the unique predictable FV process, starting from zero, such that {X-A} is a local martingale. However, this is considerably different from the definition of the compensator in the dicrete-time Doob decomposition (1). In this section, I will show how the continuous-time compensator does arise as the limit of discrete-time compensators along partitions. As previously discussed for approximating the compensator of integrable variation processes, we start by defining the notion of a stochastic partition, P, of {{\mathbb R}_+}. This is just a sequence of stopping times
分解（2）中出现的进程 A 被称为 X 的补偿器。根据定义，它是唯一可预测的有限变差（FV）过程，从零开始，使得 {X-A} 成为局部鞅。然而，这与离散时间 Doob 分解（1）中补偿器的定义有显著不同。在本节中，我将展示连续时间补偿器如何作为离散时间补偿器沿分割的极限出现。正如之前讨论的，为了近似可积变差过程的补偿器，我们首先定义 {{\mathbb R}_+} 的随机分割 P 的概念。这仅仅是一系列停时的序列。

\displaystyle 0=\tau_0\le\tau_1\le\tau_2\le\cdots\uparrow\infty.

The mesh of the partition is denoted by {\vert P\vert=\sup_n(\tau_n-\tau_{n-1})}. For a process X, we calculate the compensator along the partition P as the process {A^P_t} defined by
分区的网格用 {\vert P\vert=\sup_n(\tau_n-\tau_{n-1})} 表示。对于过程 X，我们沿着分区 P 计算补偿器，定义为过程 {A^P_t} 。

\displaystyle A^P_t=\sum_{n=1}^\infty1_{\{\tau_{n-1} < t\}}{\mathbb E}\left[X_{\tau_n}-X_{\tau_{n-1}}\;\vert\mathcal{F}_{\tau_{n-1}}\right]. (9)

Here, we are only going to consider processes X which are submartingales of class (D). This ensures that the limit {X_\infty=\lim_{t\rightarrow\infty}X_t} exists, and that {X_\tau} is integrable for all stopping times {\tau}. So, the expectations in (9) are well defined.
在这里，我们仅考虑属于(D)类下鞅的过程 X。这确保了极限 {X_\infty=\lim_{t\rightarrow\infty}X_t} 存在，且对于所有停时 {\tau} ， {X_\tau} 是可积的。因此，(9)式中的期望是明确定义的。

If X is a class (D) submartingale, then Theorem 1 says that {X=M+A} for a uniformly integrable martingale M and predictable increasing process A starting from zero and such that {A_\infty} is integrable. Therefore, by optional sampling, we have {M_\tau={\mathbb E}[M_\infty\vert\mathcal{F}_\tau]} for each stopping time {\tau}. This implies that {{\mathbb E}[M_{\tau_n}-M_{\tau_{n-1}}\vert\mathcal{F}_{\tau_{n-1}}]=0}, so (9) can be rewritten as
如果 X 是一个类(D)下鞅，那么定理 1 表明，对于一个一致可积鞅 M 和从零开始的可料递增过程 A，且使得 {A_\infty} 可积，有 {X=M+A} 。因此，通过可选抽样，对于每个停时 {\tau} ，我们有 {M_\tau={\mathbb E}[M_\infty\vert\mathcal{F}_\tau]} 。这意味着 {{\mathbb E}[M_{\tau_n}-M_{\tau_{n-1}}\vert\mathcal{F}_{\tau_{n-1}}]=0} ，因此(9)可以重写为

\displaystyle A^P_t=\sum_{n=1}^\infty1_{\{\tau_{n-1} < t\}}{\mathbb E}\left[A_{\tau_n}-A_{\tau_{n-1}}\;\vert\mathcal{F}_{\tau_{n-1}}\right]. (10)

This expresses {A^P} in terms of the integrable and increasing process A. In the case where X is quasi-left-continuous, so that A is continuous, the approximation to the compensator calculated along partitions P converges uniformly to A as the mesh goes to zero. Here, {\vert P\vert\xrightarrow{\rm P}0} denotes the limit as the mesh {\vert P\vert} goes to zero in probability.
这表达了 {A^P} 在可积且递增过程 A 中的情况。当 X 是拟左连续时，A 是连续的，沿着分区 P 计算的补偿器的近似值随着网格趋近于零而一致收敛到 A。这里， {\vert P\vert\xrightarrow{\rm P}0} 表示当网格 {\vert P\vert} 在概率上趋近于零时的极限。

Theorem 4 Let X be a cadlag and quasi-left-continuous submartingale of class (D), and A be as in decomposition (2). Then, {A^P\rightarrow A} uniformly in {L^1} as {\vert P\vert\rightarrow0} in probability. That is,
定理 4 设 X 为类(D)的右连左极且拟左连续下鞅，A 如分解(2)中所述。则， {A^P\rightarrow A} 在 {L^1} 上一致地依概率收敛于 {\vert P\vert\rightarrow0} 。即，

\displaystyle \lim_{\vert P\vert\xrightarrow{\rm P}0}{\mathbb E}\left[\sup_{t\ge0}\vert A^P_t-A_t\vert\right]=0. (11)

Proof: As X is quasi-left-continuous, its compensator A is continuous. Then, (10) expresses {A^P} in terms of the continuous process A of integrable total variation. Theorem 10 of the post on compensators then states that the limit (11) holds. ⬜
证明：由于 X 是拟左连续的，其补偿器 A 是连续的。因此，(10)式用可积全变差的连续过程 A 表达了 {A^P} 。随后，关于补偿器的帖子中的定理 10 指出，极限(11)成立。⬜

In the case where X is not quasi-left-continuous, then convergence to the compensator is not guaranteed in such a strong sense as above. As was previously shown by an example, even in the very simple case where {X=1_{[\tau,\infty)}} for some stopping time {\tau}, it is not guaranteed that the limit of {A^P_t} exists in probability as the mesh goes to zero. Instead, we have to work with respect to weak convergence in {L^1}.
在 X 不是拟左连续的情况下，不能保证补偿器会像上述那样以如此强烈的意义收敛。正如之前通过一个例子所展示的，即使在非常简单的 {X=1_{[\tau,\infty)}} 情况下，对于某个停时 {\tau} ，也不能保证当网格趋近于零时， {A^P_t} 的极限在概率上存在。相反，我们必须处理 {L^1} 中的弱收敛。

Theorem 5 Let X be a cadlag submartingale of class (D), and A be as in decomposition (2). Then, {A^P_\tau\rightarrow A_\tau} weakly in {L^1} as {\vert P\vert\rightarrow0} in probability, for any random time {\tau\colon\Omega\rightarrow{\mathbb R}_+\cup\{\infty\}}. That is,
定理 5 设 X 为(D)类的右连左极下鞅，A 如分解(2)中所述。则对于任意随机时间 {\tau\colon\Omega\rightarrow{\mathbb R}_+\cup\{\infty\}} ， {A^P_\tau\rightarrow A_\tau} 在 {L^1} 中弱收敛于 {\vert P\vert\rightarrow0} 依概率成立。即，

\displaystyle \lim_{\vert P\vert\xrightarrow{\rm P}0}{\mathbb E}\left[YA^P_\tau\right]={\mathbb E}\left[YA_\tau\right] (12)

for all uniformly bounded random variables Y.
对于所有一致有界的随机变量 Y。

Proof: As in the proof of Theorem 4, we note that (10) expresses {A^P} in terms of the integrable variation process A. Theorem 11 of the post on compensators states that the limit (12) holds. ⬜
证明：如定理 4 的证明中所述，我们注意到(10)式将 {A^P} 表示为可积变差过程 A 的函数。关于补偿器的帖子中的定理 11 指出，极限(12)成立。⬜

Notes   注释

As discussed above, the approach used to prove the Doob-Meyer decomposition in this post is quite different from the methods which are often used elsewhere. This is because we have already developed much of the theory of semimartingales independently of the Doob-Meyer decomposition, so it seems natural to simply apply this theory here rather than starting from scratch. I will, however, briefly outline some alternative approaches which are sometimes used.
如上所述，本文中用于证明 Doob-Meyer 分解的方法与通常在其他地方使用的方法大不相同。这是因为我们已经独立于 Doob-Meyer 分解发展了半鞅理论的许多内容，因此在这里直接应用这一理论似乎比从头开始更为自然。不过，我将简要概述一些有时使用的替代方法。

One method of proving the existence of the Doob-Meyer decomposition is via the Doléans measure of a class (D) cadlag submartingale X. This is a finite measure on the predictable σ-algebra {\mathcal{P}} satisfying
证明 Doob-Meyer 分解存在的一种方法是通过类(D) cadlag 下鞅 X 的 Doléans 测度。这是在可预测σ-代数 {\mathcal{P}} 上满足的有限测度。

\displaystyle \mu(\xi)={\mathbb E}\left[\int_0^\infty\xi\,dX\right]

for all bounded elementary processes {\xi}. Proving that the function {\mu} does extend to a measure can be done by a relatively straightforward application of the Caratheodory extension theorem. Next, the Doléans measure is used to define the following finite measure on the underlying measurable space {(\Omega,\mathcal{F}_t)}. Given any bounded {\mathcal{F}_t}-measurable random variable Y, choose a cadlag modification of the {\mathcal{F}_{\cdot+}}-martingale {M_t={\mathbb E}[Y\vert\mathcal{F}_{t+}]}. Then, letting {M_{t-}=\lim_{s\uparrow\uparrow t}M_s}, define {\nu_t(Y)} by
对于所有有界初等过程 {\xi} 。证明函数 {\mu} 确实可以扩展为一个测度，可以通过相对直接地应用 Caratheodory 扩展定理来完成。接下来，Doléans 测度用于定义在基础可测空间 {(\Omega,\mathcal{F}_t)} 上的以下有限测度。给定任何有界 {\mathcal{F}_t} -可测随机变量 Y，选择 {\mathcal{F}_{\cdot+}} -鞅 {M_t={\mathbb E}[Y\vert\mathcal{F}_{t+}]} 的一个右连左极修正。然后，令 {M_{t-}=\lim_{s\uparrow\uparrow t}M_s} ，定义 {\nu_t(Y)} 为

\displaystyle \nu_t(Y)=\mu(M_-).

It can be shown that {\nu_t} is absolutely continuous with respect to {{\mathbb P}}, so the Radon-Nikodym derivative
可以证明 {\nu_t} 相对于 {{\mathbb P}} 是绝对连续的，因此 Radon-Nikodym 导数

\displaystyle A_t=\frac{d\nu_t}{d{\mathbb P}}

exists. Furthermore, A is adapted, increasing and right-continuous in probability. So, it has a right-continuous modification. Also, from the construction, it is straightforward to show that {X-A} is a martingale and
存在。此外，A 是适应的、递增的且在概率上右连续。因此，它有一个右连续的修正。同样，从构造中可以直接证明 {X-A} 是一个鞅。

\displaystyle {\mathbb E}\left[M_tA_t\right]={\mathbb E}\left[\int_0^tM_{s-}\,dA_s\right] (13)

for all cadlag and bounded martingales M. The fact that A is predictable follows from (13) together with the techniques developed in the earlier post on predictable FV processes. Filling in the details, this provides a proof of the Doob-Meyer decomposition for class (D) submartingales.
对于所有右连左极且有界的鞅 M，A 的可预测性由(13)以及之前关于可预测有限变差过程的帖子中开发的技术共同得出。填补这些细节，这为(D)类下鞅的 Doob-Meyer 分解提供了证明。

The method just described, making use of the Doléans measure of X, is close to the standard `classical’ proof of the Doob-Meyer decomposition. The method is closely related to the idea of dual predictable projection, which is sometimes employed in the proof. The most difficult part is showing that any process A satisfying (13) is indeed predictable.
上述方法利用 X 的 Doléans 测度，与 Doob-Meyer 分解的标准“经典”证明相近。该方法与双重可预测投影的概念密切相关，后者有时在证明中被采用。最困难的部分在于证明任何满足(13)的过程 A 确实是可预测的。

An alternative method which is sometimes used is to construct the compensator A directly, by taking the limit of the discrete approximation along a sequence of partitions. That is, prove Theorem 5 without assuming a-priori that the compensator A exists. Again, showing that the process A is indeed predictable is often the most tricky part of this approach. Similarly, the decomposition could be constructed by first proving Theorem 4. This would prove the Doob-Meyer decomposition for quasi-left-continuous cadlag submartingales where, now, the compensator A is the unique continuous increasing process starting from zero such that {X-A} is a martingale. Extending to general cadlag submartingales is relatively straightforward. We would simply subtract out the jumps of X which occur at predictable times, and construct the compensators at these jump times explicitly.
有时使用的另一种方法是直接构造补偿器 A，通过沿着一系列分割的离散逼近取极限。也就是说，在不先验假设补偿器 A 存在的情况下证明定理 5。同样，证明过程 A 确实是可预测的通常是这种方法中最棘手的部分。类似地，分解可以通过首先证明定理 4 来构造。这将证明拟左连续右连左极下鞅的 Doob-Meyer 分解，其中补偿器 A 是从零开始的唯一连续递增过程，使得 {X-A} 是一个鞅。扩展到一般的右连左极下鞅相对简单。我们只需减去在可预测时间发生的 X 的跳跃，并在这些跳跃时间显式地构造补偿器。

Finally, as quite a lot of the theory of semimartingales and stochastic integration has already been developed in these notes, and was drawn upon in the proof given here, I think that it is worth briefly mentioning what was really needed for the proof above. The main point is that, for every submartingale X, the set (5) is bounded in probability. Furthermore, this implies that the quadratic variation {[X]} exists. Sometimes, semimartingales are defined as cadlag adapted processes such that (5) is bounded in probability. Then, quadratic variations and covariations of semimartingales always exist. This allows us to define the vector space V of semimartingales X such that {[X]_\infty} is integrable, with the inner product {\langle X,Y\rangle={\mathbb E}[[X,Y]_\infty]}. So, for any submartingale X such that {[X]_\infty} is integrable, we can define M to be the orthogonal projection in V of X onto the subspace of uniformly square integrable martingales. In these notes, this projection was done as part of the proof of the Bichteler-Dellacherie theorem. The process {A=X-M} obtained is characterized by the property that {[A,N]} is a local martingale for all local martingales N, which we showed is equivalent to A being a predictable FV process.
最后，鉴于这些笔记中已经发展了大量关于半鞅和随机积分的理论，并且在此处给出的证明中也有所引用，我认为值得简要提及上述证明真正需要的内容。关键在于，对于每一个下鞅 X，集合(5)在概率上是有界的。此外，这暗示着二次变差 {[X]} 的存在。有时，半鞅被定义为使得(5)在概率上有界的右连左极适应过程。这样一来，半鞅的二次变差和协变差总是存在的。这使我们能够定义半鞅 X 的向量空间 V，其中 {[X]_\infty} 是可积的，并带有内积 {\langle X,Y\rangle={\mathbb E}[[X,Y]_\infty]} 。因此，对于任何使得 {[X]_\infty} 可积的下鞅 X，我们可以定义 M 为 X 在 V 中向一致平方可积鞅子空间的正交投影。在这些笔记中，这一投影是作为 Bichteler-Dellacherie 定理证明的一部分完成的。所得到的 {A=X-M} 过程具有这样的特性：对于所有局部鞅 N， {[A,N]} 是一个局部鞅，我们证明了这等价于 A 是一个可预测的有限变差过程。

Published by George Lowther

Published

5 thoughts on “The Doob-Meyer Decomposition”

  1. For a recent paper on the Doob-Meyer decomposition theorem, see

    M. Beiglböck, W. Schachermayer, B. Veliyev. A short proof of the Doob-Meyer Theorem. Preprint, 2010.

    It uses Komlos lemma as in previous works of W. Schachermayer.

    Reply

    2. Ok, I’ve read it now. It seems like a very neat approach, and is something that I have thought about before. It constructs the compensator A by calculating the limit of discrete Doob decompositions along a sequence of partitions. This is (roughly) similar to proving Theorem 5 above, for a specific sequence of partitions (dyadic partitions) without a-priori assuming the existence of a compensator. One way is to use the fact that A^P_t is uniformly integrable as P runs through the partitions, which implies that this sequence is compact in the weak topology on L1 (uniformly integrable subsets of L1 are weakly relatively compact). You still need need to show that any limit point A obtained is predictable. In the paper you mention, they use the fact that by taking convex combinations, you can pass from a uniformly integrable sequence to one converging both in L1 and almost-surely (and, once you know this fact, you can pass to convex combinations without having to explicitly mention weak convergence). Then, by showing that A can be obtained almost surely as a limit of convex combinations of the left-continuous and adapted (hence, predictable) processes AP, you get that A is predictable.

      Constructing A using a limit (or limit point) of Doob-style approximations on a sequence of partitions is not particularly new — this is what Kallenberg does in Foundations of Modern Probability (iirc). The trick is in how you show that A is predictable. In the linked paper, Komlos’ lemma allows you to show that it is a limit of convex combinations of the predictable processes AP, which establishes predictability in a particularly simple way.

      Thanks for bringing this paper to my attention.

      Reply

      1. I have a question regarding the paper by M. Beiglböck, W. Schachermayer, B. Veliyev. on the Doob-Meyer decomposition which you discussed above (see previous comments). The proof states that M_t=E[M|F_t] is a cadlag martingale but my understanding is that there is a version/modification which is a cadlag martingale.
        I was wondering if this matters at all in the proof? I have little experience with this topic and so am unaware of the type of problems that can arise in the constructions of the type that this paper uses and am not sure how much difference the modification procedure would make. Thanks for for your time and hopefully this comment is not too confusing.

        Reply

        1. Actually, I think the argument is assuming that you take a cadlag version. This is required when they show that A is right continuous, by evaluating at a stopping time, which only works (using optional sampling) if you use cadlag versions of the martingale so.

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