Continuous-time filtrations and all-pairs martingales

Assume the Axiom of Choice. On a probability space $(\mathrm{\Omega },\mathcal{F},P)$, a continuous-time filtration is a family $({\mathcal{F}}_t{)}_{t\ge 0}$ of sub-sigma-algebras of $\mathcal{F}$ such that ${\mathcal{F}}_s\subseteq {\mathcal{F}}_t$ whenever $0\le s\le t$. A process $X=(X_t{)}_{t\ge 0}$ is adapted when $X_t$ is measurable from $(\mathrm{\Omega },{\mathcal{F}}_t)$ to its state space at every $t$.

For any process of random elements $X=(X_t{)}_{t\ge 0}$, its natural filtration is

${\mathcal{F}}_t^X=\sigma \text{\hspace{0.17em}⁣}(\{X_u^{-1}(C):0\le u\le t,C\text{ measurable in the state space of }{X}_u\})\mathrm{.}$

The generated sigma-algebra exists by Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal. Its generator families are nested in $t$, so $({\mathcal{F}}_t^X{)}_{t\ge 0}$ is a filtration; every $X_t$ is ${\mathcal{F}}_t^X$-measurable, and minimality makes this the smallest filtration to which $X$ is adapted. No completion or right-continuous augmentation is included.

A real process $M=(M_t{)}_{t\ge 0}$ is an all-pairs continuous-time martingale relative to $({\mathcal{F}}_t)$ when:

1. $M$ is adapted;
2. $E\mathrm{\mid }{M}_t\mathrm{\mid }<\mathrm{\infty }$ for every $t\ge 0$; and
3. for every $0\le s\le t$, $E[M_t\mid {\mathcal{F}}_s]=M_s\text{almost surely}\mathrm{.}$

The equality in clause 3 is equality of the almost-everywhere classes in Conditional expectation as an ae class. At $s=t$ it is the known-variable identity. The word “continuous-time” specifies the index set; it does not assert path continuity. Likewise the definition imposes neither right continuity nor completeness on the filtration. AC is declared exactly because the library's conditional-expectation existence theorem uses it; the filtration, adaptation, and natural-filtration constructions make no choices.

Sousi, Section 2 and Definition 2.1, printed pp. 13--14, gives natural filtrations, adaptation, integrability, and the all-pairs martingale identity in discrete time. Section 3.1, printed pp. 28 and 33, replaces the index set by ${\mathbb{R}}_{+}$, defines continuous-time filtrations and adaptation, and states that the martingale definition is unchanged. The nonaugmentation and almost-everywhere-class conventions are made explicit here to match the library's conditional-expectation interface.