Square-integrable Brownian terminal variables have Ito representations

Given: AC, a standard Brownian motion $B$ with usual augmented filtration $({\mathcal{F}}_t)$, a horizon $T>0$, and $X\in L^2({\mathcal{F}}_T)$.

Representation theorem, $L^2$ clause. For every $Z\in L^2({\mathcal{F}}_T)$ there is a predictable $H$ with finite energy on $[0,T]$ such that $Z=EZ+{\int }_0^{T}HdB$ almost surely; the conditional-expectation martingale $E[Z\mid {\mathcal{F}}_t]$ agrees up to indistinguishability with $EZ+{\int }_0^{t}HdB$, and $H$ is unique modulo $(\mathrm{d}t\otimes P)$-null sets. Brownian-filtration martingale representation

Isometry and martingale property. For finite-energy predictable $H$ the integral ${\int }_0^{T}HdB$ has mean zero and $L^2$ norm squared $E{\int }_0^T{H}^{2}ds$, and the process $t\mapsto {\int }_0^{t}HdB$ has a continuous version that is a martingale; if two finite-energy integrands have integrals with the same terminal value almost surely, their difference has zero $L^2(\mathrm{d}t\otimes P)$ norm. Ito isometry and linearity in predictable L2 The Ito integral process has a continuous martingale version Ito integral for square-integrable predictable processes Locally square-integrable predictable Brownian integrands

Conditional expectation. $E[X\mid {\mathcal{F}}_t]$ is the unique a.s. class with ${\int }_{A}E[X\mid {\mathcal{F}}_t]dP={\int }_{A}XdP$ for all $A\in {\mathcal{F}}_t$, and the tower property identifies $E[X\mid {\mathcal{F}}_T]=X$ as an a.s. class. Conditional expectation as an ae class Tower property of conditional expectation Continuous-time adapted processes and martingales

AC bookkeeping. Choice is an ambient assumption, not a source of Brownian or conditional-expectation data. It is declared because the representation theorem [F1], the Ito construction and martingale interfaces [F2], and the conditional-expectation interfaces [F3] are themselves stated under AC. The Axiom of Choice