Elementary predictable Brownian integrands

Assume the Axiom of Choice The Axiom of Choice, and fix a probability space $(\mathrm{\Omega },\mathcal{F},P)$ with a continuous-time filtration $({\mathcal{F}}_t{)}_{t\ge 0}$ Continuous-time filtrations and all-pairs martingales and a standard Brownian motion $B$ Brownian motion on it. Standing hypothesis (H). $B$ is adapted to $({\mathcal{F}}_t)$ and for all $0\le s<t$ the increment $B_t-B_s$ is independent of ${\mathcal{F}}_s$ and has law $N(0,t-s)$. The raw natural filtration and the usual augmented natural filtration Natural and usual augmented Brownian filtrations both satisfy (H) for a standard Brownian motion, by Future-path Markov property; for a general filtration, (H) is part of the data and is not automatic. Every statement in this development names (H) when it is used.

Throughout, fix a finite horizon $T>0$. An elementary predictable Brownian integrand on $[0,T]$ is a process of the form $H_s(\omega )={\sum }_{k=0}^{m-1}{\xi }_k(\omega )1_{(t_k,t_{k+1}]}(s),s\in [0,T],$ where $0=t_0<t_1<\cdots <t_m=T$ is a finite partition of $[0,T]$, each ${\xi }_k$ is a bounded real ${\mathcal{F}}_{t_k}$-measurable random variable, and the values at the partition points are irrelevant because the intervals are left-open and right-closed. The mesh of the representation is ${\max }_k(t_{k+1}-t_k)$, and the supremum norm of the representation is ${\max }_k\mathrm{\parallel }{\xi }_k{\mathrm{\parallel }}_{\mathrm{\infty }}$. The integrand itself is the process $H$; a second list of the same name with different coefficients is the same elementary integrand only if the two processes coincide in the almost-everywhere sense made precise below.

The following properties are part of the definition and are used at once.

1. Predictability. $H$ is predictable Progressively measurable and predictable processes. Indeed, for a Borel set $\mathrm{\Gamma }\subseteq \mathbb{R}$, $\{H\in \mathrm{\Gamma }\}=Z_{\mathrm{\Gamma }}\cup {\bigcup }_{k=0}^{m-1}((t_k,t_{k+1}]\times \{{\xi }_k\in \mathrm{\Gamma }\}),Z_{\mathrm{\Gamma }}:=\{\begin{array}{ll}\{0\}\times \mathrm{\Omega },& 0\in \mathrm{\Gamma },\\ \mathrm{\varnothing },& 0\mathrm{\notin }\mathrm{\Gamma },\end{array}$ a finite union of generators of the predictable sigma-algebra because $\{{\xi }_k\in \mathrm{\Gamma }\}\in {\mathcal{F}}_{t_k}$ and $\{0\}\times \mathrm{\Omega }$ is a time-zero generator. Consequently $H$ is progressively measurable and measurable for the product sigma-algebra $\mathcal{B}([0,T])\otimes \mathcal{F}$, and $H$ belongs to $L^2([0,T]\times \mathrm{\Omega },\mathrm{d}t\otimes P)$: with $c:={\max }_k\mathrm{\parallel }{\xi }_k{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$ one has $E{\int }_0^T{H}_s^{2}ds\le c^{2}T<\mathrm{\infty }$.

2. Endpoint and null-set conventions. Replacing the intervals by $(t_k,t_{k+1}]$ makes the value at $s=0$ zero for every elementary integrand. More generally, if two elementary integrands agree for all $s\in [0,T]$ except at finitely many deterministic times, then they agree $(\mathrm{d}t\otimes P)$-almost everywhere, since a finite set of times is Lebesgue-null and Tonelli computes $\mathrm{d}t\otimes P(\{u\}\times \mathrm{\Omega })=0$. All integrands and all integrals below are therefore elements of the quotient spaces of $L^2(\mathrm{d}t\otimes P)$ and $L^2(P)$; a claim about a process is a claim about its almost-everywhere class unless a representative is explicitly named, and path statements name the continuous representative.

3. Deterministic coefficients. If every ${\xi }_k$ is a deterministic real number, $H$ is a deterministic step function on $[0,T]$, so the elementary integrands include all step functions with deterministic coefficients. These are the integrands for which the integral is a Gaussian variable below.

The Axiom of Choice is declared because the Brownian construction and the conditional-expectation interface used in items 6, 7 and 13 assume it; the definition itself, including the predictability computation of clause 1, uses no choice. The countable-choice obligations inherited from that interface are declared as dependencies of this item.

The Axiom of Choice

Continuous-time filtrations and all-pairs martingales

Brownian motion

Natural and usual augmented Brownian filtrations

Future-path Markov property

Progressively measurable and predictable processes

For a fixed $\omega$, the picture illustrates a representation with three intervals. The heights ${\xi }_k(\omega )$ are schematic; no ordering or positivity of the coefficients is assumed. An open circle excludes the left endpoint of a step, and a filled circle includes its right endpoint. The separate filled point at the origin records $H_0(\omega )=0$.

Here each ${\xi }_k$ is bounded and ${\mathcal{F}}_{t_k}$-measurable: the coefficient on $(t_k,t_{k+1}]$ uses information available at its left endpoint. At an interior partition time $t_k$ the displayed representative takes the preceding step's value ${\xi }_{k-1}(\omega )$. The mesh is the largest horizontal interval length ${\max }_{0\le k<3}(t_{k+1}-t_k)$, independent of the depicted heights. The process is still identified up to $\mathrm{d}t\otimes P$-almost-everywhere equality as in the definition; the picture does not change that convention or replace the predictability argument.