Doob-Meyer Decomposition¶
By now we know what martingales look like (Martingales), when they converge (Martingale Convergence), and why Uniform Integrability is the sharp condition for the \(L^1\) theory. The Doob-Meyer decomposition completes this structural picture: every "reasonable" adapted process splits into a pure-noise martingale and a predictable drift. Itô processes are a special case; the theorem says the structure is universal.
Discrete Time: The Elementary Case¶
The discrete decomposition requires no heavy machinery.
Discrete Doob decomposition
Let \((X_n, \mathcal{F}_n)_{n\ge 0}\) be an adapted integrable process. There exist unique processes \(M\) (martingale, \(M_0 = 0\)) and \(A\) (predictable, \(A_0 = 0\)) with
If \(X\) is a submartingale, \(A\) is non-decreasing.
Construction. Set \(A_n - A_{n-1} = \mathbb{E}[X_n - X_{n-1} \mid \mathcal{F}_{n-1}]\); this is \(\mathcal{F}_{n-1}\)-measurable, so \(A\) is predictable. Define \(M_n = X_n - X_0 - A_n\); by design \(\mathbb{E}[M_n - M_{n-1} \mid \mathcal{F}_{n-1}] = 0\).
Uniqueness. Any difference \(D_n\) of two such decompositions is both a predictable process and a martingale with \(D_0 = 0\). Predictability means \(D_n\) is \(\mathcal{F}_{n-1}\)-measurable, so \(\mathbb{E}[D_n - D_{n-1} \mid \mathcal{F}_{n-1}] = D_n - D_{n-1}\). The martingale property also makes this \(0\), so \(D_n \equiv 0\). \(\square\)
Reading the pieces
\(A_n\) is the conditional drift accumulated through time \(n\). \(M_n\) is what remains after this drift is removed — the "unanticipated" part of \(X_n\).
Continuous Time: Class (D) Submartingales¶
In continuous time, two refinements are needed: paths should be càdlàg, and the stopped family \(\{X_\tau : \tau \text{ bounded stopping time}\}\) should be uniformly integrable. The latter is class (D); it rules out explosions under random sampling. A UI martingale is class (D); \(W_t^2\) is not, but \(W_{t\wedge T}^2\) is.
Doob-Meyer decomposition
Every càdlàg submartingale \((X_t)\) of class (D) admits a unique decomposition
with \(M\) a càdlàg martingale and \(A\) a predictable, non-decreasing, càdlàg process with \(A_0 = 0\).
The process \(A\) is the compensator (or dual predictable projection) of \(X\): the best predictable drift one can strip away. Without predictability, uniqueness fails — predictability is the essential regularity.
Idea of proof. Discretize time along a fine partition, apply the discrete Doob decomposition, and pass to the limit; class (D) provides the uniform integrability needed for the compensators to converge, and properties of the predictable \(\sigma\)-algebra identify the limit. Uniqueness reduces to: a predictable local martingale of finite variation starting at \(0\) is identically zero. (In the continuous case this is immediate — such a process has zero quadratic variation.) Details are carried out in Dellacherie-Meyer or Revuz-Yor.
Key Examples¶
Squared Brownian motion
By Itô's formula, \(W_t^2 = 2\int_0^t W_s\,dW_s + t\). The decomposition reads
The compensator \(A_t = t\) equals the quadratic variation \([W]_t\).
Convex transform of a martingale
If \(M\) is a continuous martingale and \(f \in C^2\) is convex, Itô's formula gives
Convexity (\(f'' \ge 0\)) makes \(A_t\) non-decreasing, confirming that \(f(M_t)\) is a submartingale.
Absolute value and local time
\(|W_t|\) is a submartingale. Tanaka's formula gives its decomposition
where \(L_t^0\) is the local time of \(W\) at \(0\) — a continuous, non-decreasing process that grows only when \(W_t = 0\).
Compensators and Quadratic Variation¶
For a continuous local martingale \(M\), the quadratic variation \([M]_t\) is the unique continuous increasing process such that \(M_t^2 - [M]_t\) is a local martingale. Comparing with Doob-Meyer:
For Brownian motion, \([W]_t = t\), recovering \(W_t^2 - t\) is a martingale.
Semimartingales¶
Allowing \(A\) to be any adapted càdlàg finite-variation process (not necessarily predictable or increasing) and \(M\) to be a local martingale defines the class of semimartingales:
Every class (D) submartingale is a semimartingale by Doob-Meyer, and Itô processes are the prime examples. Semimartingales are the natural integrators for stochastic integration, and the class is preserved under \(C^2\) transformations via Itô's formula.
Why It Matters¶
A process is a martingale exactly when its Doob-Meyer compensator vanishes, so the decomposition is the standard tool for testing and for change of measure: Girsanov shifts the compensator, and the risk-neutral measure is the one under which the compensator of discounted prices is zero. This is why the decomposition sits at the foundation of pricing theory.
Exercises¶
Exercise 1. Let \(X_n = \sum_{k=1}^n Y_k\) with \(Y_k \ge 0\) and \(\mathbb{E}[Y_k \mid \mathcal{F}_{k-1}] = c > 0\). Find the Doob decomposition and identify it as a sub- or supermartingale.
Solution to Exercise 1
\(A_n - A_{n-1} = \mathbb{E}[X_n - X_{n-1} \mid \mathcal{F}_{n-1}] = \mathbb{E}[Y_n \mid \mathcal{F}_{n-1}] = c\), so \(A_n = cn\) (predictable and strictly increasing). The martingale part is \(M_n = X_n - cn = \sum_{k=1}^n (Y_k - c)\). Since \(A_n\) is increasing, \(X_n\) is a submartingale. \(\square\)
Exercise 2. Use Itô's formula to find the Doob-Meyer decomposition of \(f(W_t)\) for \(f \in C^2\) convex. Verify on \(f(x) = x^2\) and \(f(x) = e^{\theta x}\).
Solution to Exercise 2
Itô's formula gives
so \(M_t = \int_0^t f'(W_s)\,dW_s\) (local martingale) and \(A_t = \tfrac{1}{2}\int_0^t f''(W_s)\,ds\) (non-decreasing by convexity).
For \(f(x) = x^2\): \(M_t = 2\int_0^t W_s\,dW_s\), \(A_t = t\). For \(f(x) = e^{\theta x}\): \(M_t = \theta\int_0^t e^{\theta W_s}\,dW_s\), \(A_t = \tfrac{\theta^2}{2}\int_0^t e^{\theta W_s}\,ds\). \(\square\)
Exercise 3. State Tanaka's formula for \(|W_t|\) and identify the martingale and compensator parts.
Solution to Exercise 3
Tanaka's formula:
where \(L_t^0\) is local time at \(0\). The martingale part is \(M_t = \int_0^t \operatorname{sgn}(W_s)\,dW_s\) (a true martingale, since \(|\operatorname{sgn}| \le 1\) gives \(L^2\) bound). The compensator is \(A_t = L_t^0\): continuous, non-decreasing, grows only on \(\{W_t = 0\}\). Classical Itô fails here because \(|x|\) is not \(C^2\) at \(0\); the local time is precisely what records the "kink." \(\square\)
Exercise 4. Show uniqueness of the discrete Doob decomposition: a predictable martingale starting at \(0\) is identically zero.
Solution to Exercise 4
Let \(D\) be predictable and a martingale with \(D_0 = 0\). Predictability makes \(D_n\) \(\mathcal{F}_{n-1}\)-measurable, so
The martingale property makes the left side \(0\). Hence \(D_n = D_{n-1}\) a.s., and induction from \(D_0 = 0\) gives \(D_n = 0\) a.s. for all \(n\). \(\square\)
Exercise 5. Why is the compensator of a continuous martingale \(M\) applied to \(M^2\) equal to \([M]\)? Illustrate on \(M = W\).
Solution to Exercise 5
Itô's formula for \(f(x) = x^2\) on the continuous local martingale \(M\) gives
so \(M_t^2 - [M]_t\) is a local martingale — the Doob-Meyer martingale part — and \([M]_t\) is the compensator. For \(M = W\), \([W]_t = t\), recovering \(W_t^2 - t\) as a martingale. \(\square\)