Compensator and doob decomposition

Assume AC The Axiom of Choice for the conditional classes in the martingale definition. For an integrable adapted real $X$ Adapted and integrable stochastic process, a Doob decomposition is $X_n=M_n+A_n\text{a.s. for each }n\ge 0,$ where $M$ is a martingale Martingale submartingale and supermartingale, $A_n$ is integrable at every time, $A_n$ is ${\mathcal{F}}_{n-1}$-measurable for $n\ge 1$ Predictable discrete time process, and $A_0=0$. The process $A$ is its compensator. In particular $M_0=X_0$ a.s.

This defines the requirements on a decomposition; existence and uniqueness require proof. The normalization fixes the possible transfer of an integrable ${\mathcal{F}}_0$-measurable variable between $M$ and $A$: such a transfer would change $A_0$. No monotonicity of $A$ is included for a general integrable adapted $X$. The AC assumption is inherited from conditional-expectation existence, and this definition selects no versions.