Progressively measurable and predictable processes

Assume the Axiom of Choice The Axiom of Choice. Let $(\mathrm{\Omega },\mathcal{F},P)$ be a probability space with a continuous-time filtration $({\mathcal{F}}_t{)}_{t\ge 0}$ Continuous-time filtrations and all-pairs martingales. All processes here are real-valued and indexed by $[0,\mathrm{\infty })$. AC is inherited from the cited continuous-time-filtration interface, whose separate martingale clause uses conditional expectation. Once the filtered probability space and stopping times are given, the progressive and predictable constructions below make no additional choice: they use only generated sigma-algebras and product rectangles.

1. Progressively measurable. A process $X=(X_t{)}_{t\ge 0}$ is progressively measurable relative to $({\mathcal{F}}_t)$ when for every $t\ge 0$ the restricted map $[0,t]\times \mathrm{\Omega }\to \mathbb{R}$, $(s,\omega )\mapsto X_s(\omega )$, is measurable for the product sigma-algebra $\mathcal{B}([0,t])\otimes {\mathcal{F}}_t$ The product sigma-algebra and its finite iterates.

2. Time-zero and interval generators. On the product space $[0,\mathrm{\infty })\times \mathrm{\Omega }$, let $\mathcal{P}$ be the sigma-algebra generated by the family $\mathcal{G}:=\{(s,u]\times A:0\le s<u<\mathrm{\infty },A\in {\mathcal{F}}_s\}\cup \{\{0\}\times A:A\in {\mathcal{F}}_0\}\mathrm{.}$ The interval generators with $s=0$ are included, so $(0,u]\times A$ with $A\in {\mathcal{F}}_0$ is a generator. The sigma-algebra $\mathcal{P}$ exists by Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal and is called the predictable sigma-algebra. A process $H=(H_t{)}_{t\ge 0}$ is predictable when the map $(s,\omega )\mapsto H_s(\omega )$ is $\mathcal{P}$-measurable. For a finite horizon $T>0$ we write ${\mathcal{P}}_T$ for the sigma-algebra on $[0,T]\times \mathrm{\Omega }$ generated by the same family restricted to $u\le T$ together with the time-zero generators; this is the trace of $\mathcal{P}$ on $[0,T]\times \mathrm{\Omega }$. Indeed, the trace of an interval generator is empty when $s\ge T$, and otherwise is $(s,\min (u,T)]\times A$, while time-zero generators are unchanged. Conversely every listed finite-horizon generator is such a trace.

3. Sections, adaptedness, and null sets. Two structural conventions are used repeatedly and are part of the definition. (a) Every generator is a measurable rectangle of $\mathcal{B}([0,\mathrm{\infty }))\otimes \mathcal{F}$, hence $\mathcal{P}\subseteq \mathcal{B}([0,\mathrm{\infty }))\otimes \mathcal{F}$; the time-zero generator $\{0\}\times A$ has $dt\otimes P$-measure zero for every $A$. A set in $\mathcal{P}$ has its section at each fixed time $s$ in ${\mathcal{F}}_s$: sets with that section property form a sigma-algebra, and each generator has the property by the increasing-filtration condition. Its section at each fixed $\omega$ is Borel in time, by the same sigma-algebra argument. (b) If a process is predictable then it is progressively measurable: for $s\le t$ the section $\omega \mapsto H_s(\omega )$ is ${\mathcal{F}}_s$-measurable $\subseteq {\mathcal{F}}_t$-measurable, and the progressive-measurability claim is the restriction of the joint map to $[0,t]\times \mathrm{\Omega }$; the restriction of a $\mathcal{P}$-measurable map to $[0,t]\times \mathrm{\Omega }$ is measurable for $\mathcal{B}([0,t])\otimes {\mathcal{F}}_t$ because every generator with $u\le t$ lies in that product sigma-algebra and the generators with $u>t$ have intersection with $[0,t]\times \mathrm{\Omega }$ of the form $(s,t]\times A$ with $s<t$ and $A\in {\mathcal{F}}_s$, or the empty set when $s\ge t$. The time-zero generators also lie in $\mathcal{B}([0,t])\otimes {\mathcal{F}}_t$.

The family $\mathcal{G}$ need not itself contain the whole space, but a finite intersection of generators is empty or again a generator, or a time-zero generator, because $(s,u]\cap (s^{\mathrm{\prime }},u^{\mathrm{\prime }}]=(s\vee s^{\mathrm{\prime }},u\wedge u^{\mathrm{\prime }}]$ and $A\cap A^{\mathrm{\prime }}\in {\mathcal{F}}_{s\vee s^{\mathrm{\prime }}}$ whenever $A\in {\mathcal{F}}_s$, $A^{\mathrm{\prime }}\in {\mathcal{F}}_{s^{\mathrm{\prime }}}$, while $\{0\}\times A$ is disjoint from every interval generator, including $(0,u]\times A^{\mathrm{\prime }}$. Two time-zero generators intersect in $\{0\}\times (A\cap A^{\mathrm{\prime }})$ with $A\cap A^{\mathrm{\prime }}\in {\mathcal{F}}_0$. Thus the family consisting of the whole space together with finite intersections of generators is a pi-system generating $\mathcal{P}$; this is the structural fact used by the density theorem below.

Because predictability is a measurability requirement for the joint map, it is preserved by pointwise limits, products and linear combinations of predictable processes. A deterministic function $h$ of time defines a predictable process exactly when $h$ is Borel measurable: the sets $C$ for which $C\times \mathrm{\Omega }\in \mathcal{P}$ form a sigma-algebra containing the intervals $(s,u]$ and $\{0\}$, hence all Borel sets. Conversely, take any fixed $\omega \in \mathrm{\Omega }$ (a probability space is nonempty) and use the Borel time-section property from 3(a) on every inverse image of an open set. This single choice uses no choice axiom. In particular the indicators $1_{[0,u]}$ and $1_{(0,u]}$ are predictable for every deterministic $u\ge 0$, and the process $s\mapsto 1_{s\le \tau (\omega )}$ for a stopping time $\tau$ Continuous-time stopping times and stopped sigma-algebras is predictable, since $\{(s,\omega ):s>\tau (\omega )\}={\bigcup }_{q\in {\mathbb{Q}}_{>0}}(q,\mathrm{\infty })\times \{\tau <q\}={\bigcup }_{q\in {\mathbb{Q}}_{>0}}{\bigcup }_{n>q}(q,n]\times \{\tau <q\}$ is a countable union of generators: for rational $q>0$ one has $\{\tau <q\}={\bigcup }_{q^{\mathrm{\prime }}<q,q^{\mathrm{\prime }}\in \mathbb{Q}}\{\tau \le q^{\mathrm{\prime }}\}\in {\mathcal{F}}_q$. The complement of that set inside $[0,\mathrm{\infty })\times \mathrm{\Omega }$ is the event of the indicator, so $1_{[0,\tau ]}$ is predictable; its left-continuity in the time variable is the visual form of the same computation.

No completeness of the filtration and no right continuity of $({\mathcal{F}}_t)$ is used or assumed. Apart from the inherited ambient assumption just recorded, the constructions in this definition are choice-free.