Locally square-integrable predictable Brownian integrands

Assume the Axiom of Choice and the standing hypothesis (H) of Elementary predictable Brownian integrands. For this localization interface, assume in addition that $({\mathcal{F}}_t{)}_{t\ge 0}$ satisfies the usual conditions: ${\mathcal{F}}_0$ contains every subset of every $P$-null event in $\mathcal{F}$, and ${\mathcal{F}}_t={\bigcap }_{u>t}{\mathcal{F}}_u$ for every $t\ge 0$. The earlier finite-energy construction does not require these additional conditions. Let $H=(H_s{)}_{s\ge 0}$ be a predictable process Progressively measurable and predictable processes. Its energy process is $A_t(\omega ):={\int }_0^t{H}_s(\omega {)}^{2}ds,t\ge 0,$ the integral of the nonnegative function $s\mapsto H_s(\omega {)}^2$; it may be $+\mathrm{\infty }$. The process $H$ is locally square-integrable when $A_t<\mathrm{\infty }\text{ almost surely for every finite }t\ge 0.$ No uniform bound over $t$ and no bound on $E{A}_t$ is imposed; the localization below converts almost-sure local finiteness into finite energy.

The following properties are part of the definition and are used in items 16 to 18.

1. Measurability and adaptation of the energy. $A$ is well defined and adapted: for each $t$ the map $(s,\omega )\mapsto H_s(\omega {)}^2{1}_{[0,t]}(s)$ is product measurable, so Tonelli Tonelli's theorem for nonnegative measurable functions on a sigma-finite product expresses $A_t={\int }_0^{\mathrm{\infty }}{H}_s^2{1}_{[0,t]}(s)ds$ as an integral of measurable sections and, for $t^{\mathrm{\prime }}\le t$, the section computation over $[0,t]$ shows that $A_t$ is ${\mathcal{F}}_t$-measurable (the integral of a nonnegative measurable function is measurable in the parameter). Thus every level or sublevel event of $A_t$ belongs to ${\mathcal{F}}_t$. On the event $G:={\bigcap }_{m\ge 1}\{A_m<\mathrm{\infty }\},$ which has probability one, the maps $t\mapsto A_t(\omega )$ are nondecreasing, finite-valued and continuous on $[0,\mathrm{\infty })$: on each $[0,m]$ the nonnegative integrand $H(\omega {)}^2$ has finite integral, and dominated convergence on that finite interval gives continuity. Moreover $G^c$ is a null event in $\mathcal{F}$, so completeness gives $G^c\in {\mathcal{F}}_0$ and every subset of $G^c$ belongs to every ${\mathcal{F}}_t$.

2. Canonical localization times. For $n\ge 1$ put ${\sigma }_n:=\inf \{t\ge 0:A_t\ge n\},\inf \mathrm{\varnothing }:=+\mathrm{\infty },{\tau }_n:={\sigma }_n\wedge n\mathrm{.}$ Then ${\tau }_n\le n$ everywhere, the sequence $({\tau }_n)$ is nondecreasing and ${\tau }_n\uparrow \mathrm{\infty }$ almost surely: on $G$, for fixed $m$, one has ${\sigma }_n>m$ and $n>m$ for every sufficiently large integer $n>A_m$, hence ${\tau }_n>m$ eventually. Each ${\tau }_n$ is a stopping time for $({\mathcal{F}}_t)$ Continuous-time stopping times and stopped sigma-algebras. For $t\ge n$ the event $\{{\tau }_n\le t\}$ is $\mathrm{\Omega }$. For $t<n$, continuity and monotonicity give $\{{\tau }_n\le t\}\cap G=\{A_t\ge n\}\cap G\mathrm{.}$ Thus the symmetric difference of $\{{\tau }_n\le t\}$ and the ${\mathcal{F}}_t$-event $\{A_t\ge n\}$ is a subset of $G^c$ and belongs to ${\mathcal{F}}_0\subseteq {\mathcal{F}}_t$ by completeness. Hence $\{{\tau }_n\le t\}\in {\mathcal{F}}_t$.

3. The localization localizes the energy. For every $n$ and every $t\ge 0$, $A_{t\wedge {\tau }_n}\le n\text{almost surely},$ because on $G$ the process $A$ is nondecreasing, $t\wedge {\tau }_n\le {\tau }_n\le n$, and $A_{{\tau }_n}\le n$: if ${\sigma }_n<n$ then continuity gives $A_{{\sigma }_n}=n$ and ${\tau }_n={\sigma }_n$, while if ${\sigma }_n\ge n$ then ${\tau }_n=n$ and $A_n\le n$ by the definition of ${\sigma }_n$ as an infimum. Consequently $E{\int }_0^t{H}_s^2{1}_{(0,{\tau }_n]}(s)ds=E{A}_{t\wedge {\tau }_n}\le n<\mathrm{\infty },$ so $H{1}_{(0,{\tau }_n]}$ is a predictable integrand of finite energy and its $L^2$ integral exists by Ito integral for square-integrable predictable processes. The same holds for $H{1}_{[0,{\tau }_n]}$, which differs from $H{1}_{(0,{\tau }_n]}$ only at $s=0$, a null set for $\mathrm{d}t\otimes P$.

4. Predictability of the truncations. The process $1_{[0,{\tau }_n]}$ is predictable for every stopping time ${\tau }_n$, by the generator computation recorded in Progressively measurable and predictable processes, and the products $H{1}_{(0,{\tau }_n]}$ and $H{1}_{[0,{\tau }_n]}$ are therefore predictable, being products of predictable functions.

These conventions are the only sense in which the definition localizes: the times ${\tau }_n$ are canonical functions of the energy process, so no auxiliary sequence of stopping times is selected, and the constants $n$ are the natural numbers. Completeness is what makes exceptional-path discrepancies measurable; right-continuity is retained as part of the standard usual-conditions convention used by the localization sources and downstream stopping theory. AC is declared because the ambient $L^2$ integral interface assumes it; the definition of the energy process and of the times ${\tau }_n$ uses no choice beyond that interface.

Van der Vaart, Definition 5.32 and Theorem 5.36, defines stochastic integration from an actual localizing sequence of stopping times and works throughout with filtrations satisfying the usual conditions. Eberle, Remark on the usual conditions and Lemma 5.11, likewise obtains the energy hitting times on the completed right-continuous filtration. The present page therefore keeps its finite-energy construction on raw filtrations but adopts the usual conditions at the point where almost-sure local energy, continuous versions and stopping must interact.