Continuous-time adapted processes and martingales

Assume the Axiom of Choice The Axiom of Choice. Fix a probability space $(\mathrm{\Omega },\mathcal{F},P)$ with a continuous-time filtration $({\mathcal{F}}_t{)}_{t\ge 0}$ in the sense of Continuous-time filtrations and all-pairs martingales. All processes in this definition are real-valued and indexed by $[0,\mathrm{\infty })$; the filtration is neither assumed complete nor right-continuous. The Axiom of Choice is declared because the conditional-expectation classes used in clause 3 are supplied by the Radon--Nikodym interface of Conditional expectation as an ae class, which assumes it; the countable-choice obligations inherited from that interface are declared as dependencies of this item.

1. Adapted. $X=(X_t{)}_{t\ge 0}$ is adapted to $({\mathcal{F}}_t)$ when $X_t$ is ${\mathcal{F}}_t$-measurable for every $t\ge 0$. This is exactly the notion of Continuous-time filtrations and all-pairs martingales.

2. Stopped process. For a stopping time $\tau$ for $({\mathcal{F}}_t)$ Continuous-time stopping times and stopped sigma-algebras the stopped process is $X_t^{\tau }(\omega ):=X_{t\wedge \tau (\omega )}(\omega ),t\ge 0,$ with the convention $t\wedge \mathrm{\infty }:=t$, so no value $X_{\mathrm{\infty }}$ is ever required and $X_0^{\tau }=X_0$ identically. This formula defines a pathwise family; adaptedness of $X$ alone does not assert measurability or adaptedness of its stopped values. If in addition $\tau \le c$ for a deterministic constant $c$, then $X_t^{\tau }=X_{t\wedge \tau }$, and only the values of $X$ on $[0,c]$ enter. Stopping at a stopping time is not the same as replacing a process by a modification; it is a pathwise operation.

3. Martingale. $M=(M_t{)}_{t\ge 0}$ is a martingale (an all-pairs continuous-time martingale) relative to $({\mathcal{F}}_t)$ when it is adapted, $E\mathrm{\mid }{M}_t\mathrm{\mid }<\mathrm{\infty }$ for every $t\ge 0$, and for all $0\le s\le t$ $E[M_t\mid {\mathcal{F}}_s]=M_s\text{almost surely}\mathrm{.}$ The equality is an equality of the almost-everywhere classes of Conditional expectation as an ae class; equivalently, every version of the conditional expectation on the left equals $M_s$ off one null set. At $s=t$ the identity reduces to the known-variable case. The word "continuous-time" refers to the index set only and does not assert path continuity.

4. Local martingale. $X=(X_t{)}_{t\ge 0}$ is a local martingale relative to $({\mathcal{F}}_t)$ when it is adapted, $E\mathrm{\mid }{X}_0\mathrm{\mid }<\mathrm{\infty }$, and there exist stopping times $({\tau }_n{)}_{n\ge 0}$ with ${\tau }_0\le {\tau }_1\le \cdots$ and ${\tau }_n\uparrow \mathrm{\infty }$ almost surely such that for every $n\ge 0$ the stopped process $X^{{\tau }_n}-X_0=(X_{t\wedge {\tau }_n}-X_0{)}_{t\ge 0}$ is a martingale in the sense of clause 3. The sequence $({\tau }_n)$ is called a localizing sequence. Equivalently, every $X^{{\tau }_n}$ is a martingale: adaptedness makes the integrable variable $X_0$ measurable with respect to every ${\mathcal{F}}_t$, so the constant process with value $X_0$ is a martingale and may be added to, or subtracted from, each stopped process. The centering merely makes every localized process start at $0$; apart from the required integrability of $X_0$, no claim is made that $X$ is integrable at a positive deterministic time, and no claim is made that $X$ has continuous paths.

5. Path and integrability attributes. A process has continuous paths when $t\mapsto X_t(\omega )$ is continuous on $[0,\mathrm{\infty })$ for every $\omega$ in an event of probability one; the usual almost-sure path conventions of Process law, modification, and indistinguishability apply. A martingale $M$ is square-integrable when $E{M}_t^2<\mathrm{\infty }$ for every $t\ge 0$, and $L^2$-bounded when ${\sup }_{t\ge 0}E{M}_t^2<\mathrm{\infty }$. These are properties of the single process under consideration, not of its versions: a modification of a martingale need not be adapted, so every later statement names the adapted versions it uses.

The following remarks specify what follows directly from this vocabulary.

1. A martingale is a local martingale. If $M$ is a martingale, the constant sequence ${\tau }_n:=n$, $n\ge 0$, localizes it: the stopped process $M^n-M_0$ is again a martingale by the martingale identity applied at the deterministic times $s\wedge n\le t\wedge n$. The converse fails; a local martingale need not be a martingale, and no such implication is used in this development.
2. Localization after a separately justified stopping operation. Suppose $({\tau }_n)$ localizes $X$, $\sigma$ is a stopping time, $X^{\sigma }$ is adapted, and each $(X^{{\tau }_n}-X_0{)}^{\sigma }$ is a martingale. Then $({\tau }_n)$ localizes $X^{\sigma }$. Indeed it still increases to infinity, $(X^{\sigma }{)}_0=X_0$, and the pathwise identity $(X^{\sigma }{)}^{{\tau }_n}-(X^{\sigma }{)}_0=X^{\sigma \wedge {\tau }_n}-X_0=(X^{{\tau }_n}-X_0{)}^{\sigma }$ verifies exactly clause 4. The stopped-piece martingale assertion and adaptedness are hypotheses here, not consequences of the unrestricted all-pairs definition. They must be established in each application. The sequence $(\sigma \wedge {\tau }_n)$ is not a substitute for $({\tau }_n)$: its almost-sure limit is $\sigma$, which need not be infinity.

No path continuity, no right continuity of the filtration, and no completeness of the underlying probability space is imposed by this definition. Choice enters only through the conditional-expectation interface named above, whose countable-choice obligations are declared as dependencies of this item.