Ito integral of an elementary predictable process

Assume the Axiom of Choice and work under the standing hypothesis (H) of Elementary predictable Brownian integrands: a filtered probability space with a standard Brownian motion $B$ adapted to the filtration, with increments $B_t-B_s$ independent of ${\mathcal{F}}_s$ of law $N(0,t-s)$. Fix $T>0$ and an elementary predictable integrand $H_s={\sum }_{k=0}^{m-1}{\xi }_k{1}_{(t_k,t_{k+1}]}(s),0=t_0<\cdots <t_m=T,$ with bounded ${\mathcal{F}}_{t_k}$-measurable coefficients ${\xi }_k$. The Ito integral of $H$ against $B$ is the process $I_t(H):={\sum }_{k=0}^{m-1}{\xi }_k(B_{t\wedge t_{k+1}}-B_{t\wedge t_k}),0\le t\le T,$ also written ${\int }_0^t{H}_{s}d{B}_s$ or $(H\cdot B{)}_t$. The sum is finite and is evaluated with the Brownian path of the given representative; on the event where $t\mapsto B_t$ is continuous it is continuous in $t$, and $I_0(H)=0$ because $0\wedge t_k=0$ for every $k$.

Three conventions are part of the definition.

1. Adaptedness and continuity. For fixed $t$, each summand ${\xi }_k(B_{t\wedge t_{k+1}}-B_{t\wedge t_k})$ is ${\mathcal{F}}_t$-measurable: if $t<t_k$, the Brownian difference and hence the summand are zero; if $t\ge t_k$, then ${\mathcal{F}}_{t_k}\subseteq {\mathcal{F}}_t$, while $B_{t\wedge u}$ is ${\mathcal{F}}_t$-measurable for every $u$ because $t\wedge u\le t$. Thus $I(H)$ is adapted. For each fixed $\omega$ the map $t\mapsto B_{t\wedge t_{k+1}}(\omega )-B_{t\wedge t_k}(\omega )$ is continuous outside the single exceptional null set of (H) on which the Brownian path is discontinuous; the finite sum is therefore continuous on the same event.

2. Linearity on a common refinement. If $H$ and $K$ are elementary integrands and a partition refines both representations, then the defining sums of $H$, $K$, $H+K$ and $aH$ are taken over that common partition, and the finite sums give $I_t(H+K)=I_t(H)+I_t(K)$ and $I_t(aH)=a{I}_t(H)$ for real $a$ identically. In particular $I_t(H)-I_t(K)=I_t(H-K)$ for the elementary integrand $H-K$ represented on that refinement. This is a rearrangement of finitely many terms, not a limiting statement.

3. Dependence on the representation is temporary. The definition attaches $I_t(H)$ to a chosen elementary representation. Elementary Ito integrals do not depend on step representation ↗ proves that two representations that agree $(\mathrm{d}t\otimes P)$-almost everywhere produce the same random variables almost surely at each fixed time, so that $I_t(H)$ is a function of the $(\mathrm{d}t\otimes P)$-class of $H$ alone. Until then, every statement about an elementary integrand names the representation it uses.

The Axiom of Choice is declared because the standing hypothesis (H) is part of the Brownian interface of Elementary predictable Brownian integrands, which assumes it; the definition of the finite sum uses no choice. The countable-choice obligations inherited from that interface are declared as dependencies of this item.