A short proof of the Doob–Meyer theorem

Abstract

Every submartingale $S$ of class $D$ has a unique Doob–Meyer decomposition $S=M+A$, where $M$ is a martingale and $A$ is a predictable increasing process starting at 0.

We provide a short proof of the Doob–Meyer decomposition theorem. Several previously known arguments are included to keep the paper self-contained.

1. Introduction

Throughout this article we fix a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ and a right-continuous complete filtration ${\left({\mathcal{F}}_t\right)}_{0\le t\le T}$.

An adapted process ${\left(S_t\right)}_{0\le t\le T}$ is of class $D$ if the family of random variables $S_{\tau }$ where $\tau$ ranges through all stopping times is uniformly integrable [9].

The purpose of this paper is to give a short proof of the following.

Theorem 1.1 Doob–Meyer —

Let $S={\left(S_t\right)}_{0\le t\le T}$ be a càdlàg submartingale of class $D$ . Then, $S$ can be written in a unique way in the form

$$S=M+A$$ (1)

where $M$ is a martingale and $A$ is a predictable increasing process starting at 0.

Doob [4] noticed that in discrete time an integrable process $S={\left(S_n\right)}_{n=1}^{\infty }$ can be uniquely represented as the sum of a martingale $M$ and a predictable process $A$ starting at 0; in addition, the process $A$ is increasing iff $S$ is a submartingale. The continuous time analogue, Theorem 1.1, goes back to Meyer [9], [10], who introduced the class $D$ and proved that every submartingale $S={\left(S_t\right)}_{0\le t\le T}$ can be decomposed in the form (1), where $M$ is a martingale and $A$ is a natural process. The modern formulation is due to Doléans-Dade [2], [3] who obtained that an increasing process is natural iff it is predictable. Further proofs of Theorem 1.1 were given by Rao [11], Bass [1] and Jakubowski [5].

Rao works with the $\sigma (L^1,L^{\infty })$-topology and applies the Dunford–Pettis compactness criterion to obtain the continuous time decomposition as a weak-$L^1$ limit from discrete approximations. To obtain that $A$ is predictable one then invokes the theorem of Doléans-Dade.

Bass [1] gives a more elementary proof based on the dichotomy between predictable and totally inaccessible stopping times.

Jakubowski [5] proceeds as Rao, but notices that predictability of the process $A$ can also be obtained through an application of Komlos’ lemma [8].

This is also our starting point. Indeed the desired decomposition can be obtained from a trivial $L^2$-Komlos lemma, making the Dunford–Pettis criterion obsolete.

2. Proof of Theorem 1.1

The proof of uniqueness is standard and we have nothing to add here; see for instance [7, Lemma 25.11].

For the remainder of this article we work under the assumptions of Theorem 1.1 and fix $T=1$ for simplicity.¹

Denote by ${\mathcal{D}}_n$ and $\mathcal{D}$ the set of $n$-th resp. all dyadic numbers $j/2^n$ in the interval $[0,1]$. For each $n$, we consider the discrete time Doob decomposition of the sampled process $S^n={\left(S_t\right)}_{t\in {\mathcal{D}}_n}$, that is, we define $A^n,M^n$ by $A_0^n≔0$,

$$A_t^n-A_{t-1/2^n}^n≔\mathbb{E}[S_t-S_{t-1/2^n}|{\mathcal{F}}_{t-1/2^n}]\text{and}$$ (2)

$$M_t^n≔S_t-A_t^n$$ (3)

so that ${\left(M_t^n\right)}_{t\in {\mathcal{D}}_n}$ is a martingale and ${\left(A_t^n\right)}_{t\in {\mathcal{D}}_n}$ is predictable with respect to ${\left({\mathcal{F}}_t\right)}_{t\in {\mathcal{D}}_n}$.

The idea of the proof is, of course, to obtain the continuous time decomposition (1) as a limit, or rather, as an accumulation point of the processes $M^n,A^n,n\ge 1$.

Clearly, in infinite dimensional spaces a (bounded) sequence need not have a convergent subsequence. As a substitute for the Bolzano–Weierstrass Theorem we establish the Komlos-type Lemma 2.1 in Section 2.1.

In order to apply this auxiliary result, we require that the sequence ${\left(M_1^n\right)}_{n\ge 1}$ is uniformly integrable. This follows from the class $D$ assumption as shown by Rao [11]. To keep the paper self-contained, we provide a proof in Section 2.2.

Finally, in Section 2.3, we obtain the desired decomposition by passing to a limit of the discrete time versions. As the Komlos-approach guarantees convergence in a strong sense, predictability of the process $A$ follows rather directly from the predictability of the approximating processes. This idea is taken from [5].

2.1. Komlos’ lemma

Following Komlos [8],² it is sometimes possible to obtain an accumulation point of a bounded sequence in an infinite dimensional space if appropriate convex combinations are taken into account.

A particularly simple result of this kind holds true if ${\left(f_n\right)}_{n\ge 1}$ is a bounded sequence in a Hilbert space. In this case

$$A=\underset{n\ge 1}{sup}inf\{{\Vert g\Vert }_2:g\in conv\{f_n,f_{n+1},\dots \}\}$$

is finite and for each $n$ we may pick some $g_n\in conv\{f_n,f_{n+1},\dots \}$ such that ${\Vert g_n\Vert }_2\le A+1/n$. If $n$ is sufficiently large with respect to $\epsilon >0$, then ${\Vert (g_k+g_m)/2\Vert }_2>A-\epsilon$ for all $m,k\ge n$ and hence

$${\Vert g_k-g_m\Vert }_2^2=2{\Vert g_k\Vert }_2^2+2{\Vert g_m\Vert }_2^2-{\Vert g_k+g_m\Vert }_2^2\le 4{(A+\frac{1}{n})}^2-4{(A-\epsilon )}^2\text{.}$$

By completeness, ${\left(g_n\right)}_{n\ge 1}$ converges in ${\Vert .\Vert }_2$.

By a straight forward truncation procedure this Hilbertian Komlos lemma yields an $L^1$-version which we will need subsequently.³

Lemma 2.1

Let ${\left(f_n\right)}_{n\ge 1}$ be a uniformly integrable sequence of functions on a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ . Then there exist functions $g_n\in conv(f_n,f_{n+1},\dots )$ such that ${\left(g_n\right)}_{n\ge 1}$ converges in ${\Vert .\Vert }_{L^1\left(\Omega \right)}$.

Proof

For $i,n\in \mathbb{N}$ set $f_n^{\left(i\right)}≔f_n{\mathbb{1}}_{\{\left|f_n\right|\le i\}}$ such that $f_n^{\left(i\right)}\in L^2\left(\Omega \right)$.

We claim that there exist for every $n$ convex weights ${\lambda }_n^n,\dots ,{\lambda }_{N_n}^n$ such that the functions ${\lambda }_n^n{f}_n^{\left(i\right)}+\cdots +{\lambda }_{N_n}^n{f}_{N_n}^{\left(i\right)}$ converge in $L^2\left(\Omega \right)$ for every $i\in \mathbb{N}$.

To see this, one first uses the Hilbertian lemma to find convex weights ${\lambda }_n^n,\dots ,{\lambda }_{N_n}^n$ such that ${({\lambda }_n^n{f}_n^{\left(1\right)}+\cdots +{\lambda }_{N_n}^n{f}_{N_n}^{\left(1\right)})}_{n\ge 1}$ converges. In the second step, one applies the lemma to the sequence ${({\lambda }_n^n{f}_n^{\left(2\right)}+\cdots +{\lambda }_{N_n}^n{f}_{N_n}^{\left(2\right)})}_{n\ge 1}$, to obtain convex weights which work for the first two sequences. Repeating this procedure inductively we obtain sequences of convex weights which work for the first $m$ sequences. Then a standard diagonalization argument yields the claim.

By uniform integrability, ${lim}_{i\to \infty }{\Vert f_n^{\left(i\right)}-f_n\Vert }_1=0$, uniformly with respect to $n$. Hence, once again, uniformly with respect to $n$,

$$\underset{i\to \infty }{lim}{\Vert (\underset{n}{\overset{n}{\lambda }}{f}_n^{\left(i\right)}+\cdots +{\lambda }_{N_n}^n{f}_{N_n}^{\left(i\right)})-({\lambda }_n^n{f}_n+\cdots +{\lambda }_{N_n}^n{f}_{N_n})\Vert }_1=0\text{.}$$

Thus ${({\lambda }_n^n{f}_n+\cdots +{\lambda }_{N_n}^n{f}_{N_n})}_{n\ge 1}$ is a Cauchy sequence in $L^1\left(\Omega \right)$. □

2.2. Uniform integrability of the discrete approximations

Lemma 2.2 [11] —

The sequence ${\left(M_1^n\right)}_{n\ge 1}$ is uniformly integrable.

Proof

Subtracting $\mathbb{E}\left[S_1\right|{\mathcal{F}}_t]$ from $S_t$ we may assume that $S_1=0$ and $S_t\le 0$ for all $0\le t\le 1$. Then $M_1^n=-A_1^n$, and for every ${\left({\mathcal{F}}_t\right)}_{t\in {\mathcal{D}}_n}$-stopping time $\tau$

$$S_{\tau }^n=-\mathbb{E}\left[A_1^n\right|{\mathcal{F}}_{\tau }]+A_{\tau }^n\text{.}$$ (4)

We claim that ${\left(A_1^n\right)}_{n=1}^{\infty }$ is uniformly integrable. For $c>0,n\ge 1$ define

$${\tau }_n\left(c\right)=inf\{(j-1)/2^n:\underset{j/2^n}{\overset{n}{A}}>c\}\wedge 1\text{.}$$

From $A_{{\tau }_n\left(c\right)}^n\le c$ and (4) we obtain $S_{{\tau }_n\left(c\right)}\le -E\left[A_1^n\right|{\mathcal{F}}_{{\tau }_n\left(c\right)}]+c$.

Thus,

$${\int }_{\{A_1^n>c\}}{A}_1^{n}d\mathbb{P}={\int }_{\{{\tau }_n\left(c\right)<1\}}\mathbb{E}\left[A_1^n\right|{\mathcal{F}}_{{\tau }_n\left(c\right)}]d\mathbb{P}\le c\mathbb{P}[{\tau }_n\left(c\right)<1]-{\int }_{\{{\tau }_n\left(c\right)<1\}}{S}_{{\tau }_n\left(c\right)}d\mathbb{P}\text{.}$$

Note $\{{\tau }_n\left(c\right)<1\}\subseteq \{{\tau }_n\left(\frac{c}{2}\right)<1\}$, hence, by (4)

$${\int }_{\{{\tau }_n\left(\frac{c}{2}\right)<1\}}-S_{{\tau }_n\left(\frac{c}{2}\right)}d\mathbb{P}={\int }_{\{{\tau }_n\left(\frac{c}{2}\right)<1\}}{A}_1^n-A_{{\tau }_n\left(\frac{c}{2}\right)}^{n}d\mathbb{P}\ge {\int }_{\{{\tau }_n\left(c\right)<1\}}{A}_1^n-A_{{\tau }_n\left(\frac{c}{2}\right)}^{n}d\mathbb{P}\ge \frac{c}{2}\mathbb{P}[{\tau }_n\left(c\right)<1]\text{.}$$

Combining the above inequalities we obtain

$${\int }_{\{A_1^n>c\}}{A}_1^{n}d\mathbb{P}\le -2{\int }_{\{{\tau }_n\left(\frac{c}{2}\right)<1\}}{S}_{{\tau }_n\left(\frac{c}{2}\right)}d\mathbb{P}-{\int }_{\{{\tau }_n\left(c\right)<1\}}{S}_{{\tau }_n\left(c\right)}d\mathbb{P}\text{.}$$ (5)

On the other hand

$$\mathbb{P}[{\tau }_n\left(c\right)<1]=\mathbb{P}[A_1^n>c]\le \mathbb{E}\left[A_1^n\right]/c=-\mathbb{E}\left[M_1^n\right]/c=-\mathbb{E}\left[S_0\right]/c\text{,}$$

hence, as $c\to \infty ,\mathbb{P}[{\tau }_n\left(c\right)<1]$ goes to 0, uniformly in $n$. As $S$ is of class $D$, (5) implies that the sequence ${\left(A_1^n\right)}_{n\ge 1}$ is uniformly integrable and hence ${\left(M_1^n\right)}_{n\ge 1}={(S_1-A_1^n)}_{n\ge 1}$ is uniformly integrable as well. □

2.3. The limiting procedure

For each $n$, extend $M^n$ to a (càdlàg) martingale on $[0,1]$ by setting $M_t^n≔\mathbb{E}\left[M_1^n\right|{\mathcal{F}}_t]$. By Lemma 2.1, Lemma 2.2 there exist $M\in L^1\left(\Omega \right)$ and for each $n$ convex weights ${\lambda }_n^n,\dots ,{\lambda }_{N_n}^n$ such that with

$${\mathcal{M}}^n≔{\lambda }_n^n{M}^n+\cdots +{\lambda }_{N_n}^n{M}^{N_n}$$ (6)

we have ${\mathcal{M}}_1^n\to M$ in $L^1\left(\Omega \right)$. Then, by Jensen’s inequality, ${\mathcal{M}}_t^n\to M_t≔\mathbb{E}\left[M\right|{\mathcal{F}}_t]$ for all $t\in [0,1]$. For each $n\ge 1$ we extend $A^n$ to $[0,1]$ by

$$A^n≔\sum _{t\in {\mathcal{D}}_n}\underset{t}{\overset{n}{A}}{\mathbb{1}}_{(t-1/2^n,t]}$$ (7)

$$\text{and set }{\mathcal{A}}^n≔{\lambda }_n^n{A}^n+\cdots +{\lambda }_{N_n}^n{A}^{N_n}\text{,}$$ (8)

where we use the same convex weights as in (6). Then the càdlàg process

$${\left(A_t\right)}_{0\le t\le 1}≔{(S_t-M_t)}_{0\le t\le 1}$$

satisfies for every $t\in \mathcal{D}$

$${\mathcal{A}}_t^n=(S_t-{\mathcal{M}}_t^n)\to (S_t-M_t)=A_t\text{in }{L}^1\left(\Omega \right)\text{.}$$

Passing to a subsequence which we denote again by $n$, we obtain that convergence holds also almost surely. Consequently, $A$ is almost surely increasing on $\mathcal{D}$ and, by right continuity, also on $[0,1]$.

As the processes $A^n$ and ${\mathcal{A}}^n$ are left-continuous and adapted, they are predictable. To obtain that $A$ is predictable, we show that for a.e. $\omega$ and every $t\in [0,1]$

$$\underset{n}{lim\; sup}\underset{t}{\overset{n}{\mathcal{A}}}\left(\omega \right)=A_t\left(\omega \right)\text{.}$$ (9)

If $f_n,f:[0,1]\to \mathbb{R}$ are increasing functions such that $f$ is right continuous and ${lim}_n{f}_n\left(t\right)=f\left(t\right)$ for $t\in \mathcal{D}$, then

$$\underset{n}{lim\; sup}{f}_n\left(t\right)\le f\left(t\right)\text{for all }t\in [0,1]\text{ and}$$ (10)

$$\underset{n}{lim}{f}_n\left(t\right)=f\left(t\right)\text{if }f\text{ is continuous at }t\text{.}$$ (11)

Consequently, (9) can only be violated at discontinuity points of $A$. As $A$ is càdlàg, every path of $A$ can have only finitely many jumps larger than $1/k$ for $k\in \mathbb{N}$. It follows that the points of discontinuity of $A$ can be exhausted by a countable sequence of stopping times, and therefore it suffices to prove ${lim\; sup}_n{\mathcal{A}}_{\tau }^n=A_{\tau }$ for every stopping time $\tau$.

To do so, we argue along the lines of [5]. By (10), ${lim\; sup}_n{\mathcal{A}}_{\tau }^n\le A_{\tau }$ and as ${\mathcal{A}}_{\tau }^n\le {\mathcal{A}}_1^n\to A_1\text{ in }{L}^1\left(\Omega \right)$ we deduce from Fatou’s Lemma⁴

$$\underset{n}{lim\; inf}\mathbb{E}\left[\underset{\tau }{\overset{n}{A}}\right]\le \underset{n}{lim\; sup}\mathbb{E}\left[\underset{\tau }{\overset{n}{\mathcal{A}}}\right]\le \mathbb{E}\left[\underset{n}{lim\; sup}\underset{\tau }{\overset{n}{\mathcal{A}}}\right]\le \mathbb{E}\left[A_{\tau }\right]\text{.}$$

Therefore it is sufficient to show ${lim}_n\mathbb{E}\left[A_{\tau }^n\right]=\mathbb{E}\left[A_{\tau }\right]$. For $n\ge 1$ set

$${\sigma }_n≔inf\{t\in {\mathcal{D}}_n:t\ge \tau \}\text{.}$$

Then $A_{\tau }^n=A_{{\sigma }_n}^n$ and ${\sigma }_n\downarrow \tau$. Using that $S$ is of class $D$, we obtain

$$\mathbb{E}\left[A_{\tau }^n\right]=\mathbb{E}\left[A_{{\sigma }_n}^n\right]=\mathbb{E}\left[S_{{\sigma }_n}\right]-\mathbb{E}\left[M_0\right]\to \mathbb{E}\left[S_{\tau }\right]-\mathbb{E}\left[M_0\right]=\mathbb{E}\left[A_{\tau }\right]\text{.}$$