The Ito integral process has a continuous martingale version

Fix $T>0$ and choose, using [F8] on the nonempty sets supplied by [F1] with tolerance $2^{-2n}$, bounded elementary predictable $H^n$ with $\mathrm{\parallel }{H}^n-H{\mathrm{\parallel }}_{L^2(\mathrm{d}t\otimes P)}\le 2^{-2n}$; write $X_t^n:=I_t(H^n)$ for the continuous elementary process of [F2].

For every elementary predictable $J$ and every $S>0$, writing the path supremum as its measurable version on the scaled dyadic grid, $E{\sup }_{t\le S}\mathrm{\mid }{I}_t(J){\mathrm{\mid }}^2\le 4E{\int }_0^S{J}^{2}du$: on the full-measure event where $I(J)$ is continuous, [F5] identifies the supremum over $[0,S]$ with the supremum over the scaled dyadic grid, [F4] writes the latter as the increasing limit of the finite maxima over the nested grids $D_m$, [F3] bounds $E{\max }_{q\in D_m}\mathrm{\mid }{I}_q(J){\mathrm{\mid }}^2\le 4E\mathrm{\mid }{I}_S(J){\mathrm{\mid }}^2=4E{\int }_0^S{J}^{2}du$ for every $m$, and monotone convergence passes the bound to the limit.

For each $n$, $H^{n+1}-H^n$ is elementary on a common refinement and $\mathrm{\parallel }{H}^{n+1}-H^n{\mathrm{\parallel }}_{L^2(\mathrm{d}t\otimes P)}\le 2^{-2(n+1)}+2^{-2n}\le 2^{1-2n}$; applying step 1.2 to $J=H^{n+1}-H^n$ gives $E{a}_n^2\le 4(2^{1-2n}{)}^2=2^{4-4n}$ for the measurable random variable $a_n:={\sup }_{t\in D}\mathrm{\mid }{X}_t^{n+1}-X_t^n\mathrm{\mid }$, which equals the full supremum on the common event of continuity.

Consequently $E{\sum }_{n\ge 1}{2}^n{a}_n^2={\sum }_{n\ge 1}{2}^{n}E{a}_n^2\le {\sum }_{n\ge 1}{2}^{4-3n}<\mathrm{\infty }$ by monotone convergence for the nonnegative series [F4], so ${\sum }_n{2}^n{a}_n^2<\mathrm{\infty }$ almost surely; the Cauchy--Schwarz inequality ${\sum }_n{a}_n={\sum }_n(2^{n\mathrm{/}2}{a}_n)2^{-n\mathrm{/}2}\le \left({\sum }_n{2}^n{a}_n^2{)}^{1\mathrm{/}2}\right({\sum }_n{2}^{-n}{)}^{1\mathrm{/}2}$ then gives ${\sum }_n{a}_n<\mathrm{\infty }$ almost surely, on a measurable event $A_T$ of probability one, intersected with the common continuity event of the elementary processes.

On $A_T$ the sequence $X^n$ converges uniformly on $[0,T]$ to a limit; put $L_t:={\mathrm{lim\hspace{0.17em}sup}}_n{X}_t^n$ and define $M_t^{(T)}:=L_t$ when $L_t$ is finite, and $M_t^{(T)}:=0$ otherwise. This is real-valued and ${\mathcal{F}}_t$-measurable: $L_t$ is an extended-real measurable limit superior and its finite-value event belongs to ${\mathcal{F}}_t$. The resulting variable satisfies $M_t^{(T)}={\lim }_n{X}_t^n$ almost surely for every $t$; on $A_T$ the paths of $M^{(T)}$ are continuous, being uniform limits of continuous paths.

For every $t\in [0,T]$ the sequence $X_t^n=I_t(H^n)$ converges to ${\int }_0^{t}HdB$ in $L^2(P)$ by [F6] applied to the truncations $H^n{1}_{[0,t]}\to H{1}_{[0,t]}$; combined with step 4.1 and the almost-sure uniqueness of limits, $M_t^{(T)}={\int }_0^{t}HdB$ almost surely for every $t\in [0,T]$.

For every $t\in [0,T]$, $E(M_t^{(T)}{)}^2={\lim }_{n}E(X_t^n{)}^2={\lim }_{n}E{\int }_0^t(H^n{)}^{2}ds=E{\int }_0^t{H}^{2}ds<\mathrm{\infty }$, using [F6] and the $L^2$-norm continuity of the elementary integrals; hence $M^{(T)}$ is square-integrable.

For $0\le s\le t\le T$ and $n$, $E[X_t^n\mid {\mathcal{F}}_s]=X_s^n$ by [F2], and step 5.1 and [F7] give $X_t^n\to M_t^{(T)}$ and $X_s^n\to M_s^{(T)}$ in $L^1(P)$ as well, so [F7] lets the conditional expectations pass to the limit: $E[M_t^{(T)}\mid {\mathcal{F}}_s]=M_s^{(T)}$ almost surely.

Steps 4.1, 5.1, 6.1 and 5.2 show that $M^{(T)}$ is an adapted continuous square-integrable martingale version on $[0,T]$. If $N$ is another continuous version on $[0,T]$, then $M_q^{(T)}=N_q$ almost surely for each rational $q\in [0,T]$; intersecting the countably many measurable full-measure events and the two continuity events, and then invoking continuity, gives $M_t^{(T)}=N_t$ for every $t\in [0,T]$ on one measurable event of probability one.

Apply the construction on the horizons $T=m$, $m=1,2,\dots$; two versions on adjacent horizons agree on the smaller one by the uniqueness of step 7.1 applied there, since the restriction of the larger-horizon version is a continuous version on the smaller horizon. For $t\ge 0$ put $m(t):=\max (1,\lceil t\rceil )$ and define $M_t:=M_t^{(m(t))}$. This is well typed and ${\mathcal{F}}_t$-measurable. On the intersection of the countably many overlap-agreement and continuity events, it coincides at every time with the compatible local versions, so its path is continuous. At each deterministic time it is a version of ${\int }_0^{t}HdB$; hence the local martingale identities show that $M$ is a square-integrable martingale with $E{M}_t^2=E{\int }_0^t{H}^{2}ds$. Uniqueness on one measurable full-measure event over $[0,\mathrm{\infty })$ follows from the same rationals-and-continuity argument. AC supplies the approximants of step 1.1 simultaneously for all integer horizons, and is also inherited through the declared ambient interfaces.