Doob–Meyer decomposition theorem

Remarks

Here a process is predictable if it is measurable for the σ-algebra on $[0, \infty) \times \Omega$ generated by the left-continuous adapted processes. One proof applies the discrete Doob decomposition theorem on dyadic time grids and shows that the predictable parts converge, using class (D) for uniform integrability. Example: for a Wiener process $W$ and $T > 0$, the stopped process $X_t = W_{t \wedge T}^2$ is a submartingale of class (D), and its decomposition is $M_t = W_{t \wedge T}^2 - t \wedge T$, $A_t = t \wedge T$. Without class (D) a decomposition still exists locally, with $M$ a local martingale; this is used for $W_t^2 = (W_t^2 - t) + t$ on $[0, \infty)$. Predictability is what makes $A$ unique. For a Poisson process $N$ with rate $1$, both $N_t = 0 + N_t$ and $N_t = (N_t - t) + t$ split $N$ into a martingale and an increasing process. Only $A_t = t$ is predictable.

Going deeper

For a student

The theorem is the source of the predictable quadratic variation $\langle M \rangle$ of a square-integrable martingale $M$, the compensator $A$ of $M^2$. It underlies the construction of the stochastic integral with respect to martingales and the theory of semimartingales. Compensators of counting processes, such as $t$ for the Poisson process, come from the same theorem and are central in point process theory.