Predictable quadratic variation in discrete time

Assume AC The Axiom of Choice. For a real martingale $M$ Martingale submartingale and supermartingale with $E{M}_n^2<\mathrm{\infty }$ at every time, its predictable quadratic variation is $⟨M{⟩}_0=0,⟨M{⟩}_n={\sum }_{k=1}^{n}E[(M_k-M_{k-1}{)}^2\mid {\mathcal{F}}_{k-1}]\mathrm{.}$ Here conditional expectations denote the classes of Conditional expectation as an ae class. To obtain a process of versions, note first that $D_k=M_k-M_{k-1}$ has a measurable square Arithmetic and lattice operations preserve measurability whenever they are defined and $D_k^2\le 2M_k^2+2M_{k-1}^2,E{D}_k^2<\mathrm{\infty }\mathrm{.}$ Choose a finite real integrable ${\mathcal{F}}_{k-1}$-measurable version $b_k$ of $E[D_k^2\mid {\mathcal{F}}_{k-1}]$ for each $k$. Conditional positivity Basic algebra and order properties of conditional expectation gives $b_k\ge 0$ a.s. Replace $b_k$ by $\max (b_k,0)$: this is measurable for the same sigma-algebra and changes it only on its own measurable null set. Use these nonnegative versions in the finite sum.

For the finite-integral interface, augment every finite disjoint display of a nonnegative simple function by the complement with coefficient $0$. Intersections of two augmented displays partition the space and carry equal coefficients wherever nonempty, so finite additivity and $0\cdot (+\mathrm{\infty })=0$ prove representation independence. Common refinements give simple addition and monotonicity; scalar zero is direct and positive scalars are termwise. Taking suprema over simple minorants, using increasing simple approximations and the sets $\{f_j\ge cs\}$ for $0<c<1$, gives monotone convergence and nonnegative additivity. Positive/negative and real/imaginary decompositions give finite $L^1$ linearity. Substituting this repair at the foundation also validates the event-integral RN construction used by the cited conditional-expectation class and its positivity.

Every $b_k$ with $k\le n$ is ${\mathcal{F}}_{n-1}$-measurable; therefore $⟨M{⟩}_n$ is predictable Predictable discrete time process. The locally reconstructed finite linearity gives integrability. The chosen version has nonnegative increments at every point, and any other measurable versions define the same class at each time. AC is used in the supplied RN existence and in selecting the countable family of versions.

The optional quadratic sum is instead $[M{]}_0=0,[M{]}_n={\sum }_{k=1}^n(M_k-M_{k-1}{)}^2\mathrm{.}$ It too is integrable, adapted and increasing by the same square bound and finite-sum argument. Its summands are only required to be ${\mathcal{F}}_k$-measurable; predictability or equality to $⟨M⟩$ is not part of this definition. Neither sum includes a term $M_0^2$.