Localized Ito integral

Measurable versions and stopping: for an adapted process $X$ with almost-sure continuous paths and $X_0=0$ almost surely, define $X_0^j=0$ and $X_s^j=X_{k{2}^{-j}}$ on $(k{2}^{-j},(k+1)2^{-j}]$, $k\ge 0$. For each finite horizon these step processes are progressive by [F5]'s predictable generators and predictable-to-progressive inclusion: the coefficient is measurable at the left endpoint, so its inverse images give the required rectangles. Put ${\widehat{X}}_s={\lim }_j{X}_s^j$ where this limit exists finitely, and zero otherwise. By [F6] on each product sigma-algebra, $\widehat{X}$ is progressive. It equals $X$ at every time on the measurable full event of continuity and zero start. Thus it preserves every deterministic-time integral class and martingale identity. A progressive $Y$ has adapted stopped values: for fixed $t$, $r=t\wedge \sigma$ is ${\mathcal{F}}_t$-measurable, and the map $\omega \mapsto (r(\omega ),\omega )$ into $[0,t]\times \mathrm{\Omega }$ is measurable into $\mathcal{B}([0,t])\otimes {\mathcal{F}}_t$ by rectangle inverse images. Composition with the progressive restriction of $Y$ gives $Y_{t\wedge \sigma }\in {\mathcal{F}}_t$. Choose this construction for every finite-energy integral used below; [F7] permits the countably many required choices. All sequences indexed by positive integers are reindexed by $n=j+1$ when applying an interface indexed from zero.

Clause 4 for elementary $G$ and finite-valued $\sigma$: refine the partition of $G$ so that it contains the finitely many values of $\sigma$ and the point $t$; on each block $(t_k,t_{k+1}]$ the indicator $1_{s\le \sigma }$ is constant in $s$ with value $1_{\sigma \ge t_{k+1}}$, and $\{\sigma <t_{k+1}\}=\{\sigma \le t_k\}\in {\mathcal{F}}_{t_k}$ because $\sigma$ takes only partition values, so $G{1}_{[0,\sigma ]}{1}_{[0,t]}$ is elementary with coefficients ${\xi }_k{1}_{\sigma \ge t_{k+1}}$; its defining sum is ${\sum }_k{\xi }_k{1}_{\sigma \ge t_{k+1}}(B_{t_{k+1}\wedge t}-B_{t_k\wedge t})$, which term-by-term equals ${\sum }_k{\xi }_k(B_{(t\wedge \sigma )\wedge t_{k+1}}-B_{(t\wedge \sigma )\wedge t_k})=(G\cdot B{)}_{t\wedge \sigma }$ on the measurable full event where the progressive version agrees with the elementary sum at all times, by step 1.1.

Clause 4 for elementary $G$ and arbitrary $\sigma$: let ${\sigma }_m:=2^{-m}\lceil 2^m(\sigma \wedge m)\rceil$ be the dyadic ceiling of the bounded stopping time $\sigma \wedge m$, a finite-valued stopping time with ${\sigma }_m\ge \sigma \wedge m$ and ${\sigma }_m\to \sigma$; by step 2.1 and [F3], $(G\cdot B{)}_{t\wedge {\sigma }_m}=I_T(G{1}_{[0,{\sigma }_m]}{1}_{[0,t]})$ for every $m$.

As $m\to \mathrm{\infty }$: $(G\cdot B{)}_{t\wedge {\sigma }_m}\to (G\cdot B{)}_{t\wedge \sigma }$ in $L^2(P)$ by continuity of the path and the maximal bound [F2] with [F6] (the measurable grid supremum of [F2] supplies the dominating random variable); and $I_T(G{1}_{[0,{\sigma }_m]}{1}_{[0,t]})\to I_T(G{1}_{[0,\sigma ]}{1}_{[0,t]})$ in $L^2(P)$ because their indicators converge for Lebesgue-almost every time (the possible boundary $s=\sigma$ is irrelevant), and they are dominated by $\mathrm{\mid }G\mathrm{\mid }{1}_{[0,t]}\in L^2$, and the integral is an isometry [F4]. Hence $(G\cdot B{)}_{t\wedge \sigma }={\int }_0^{t}G{1}_{(0,\sigma ]}dB$ almost surely for elementary $G$ and every stopping time $\sigma$.

Clause 4 for general finite-energy $G$: approximate $G$ by bounded elementary $G^j$ in $L^2(\mathrm{d}t\otimes P)$ [F4]; then $(G^j\cdot B{)}_{t\wedge \sigma }\to (G\cdot B{)}_{t\wedge \sigma }$ in $L^2(P)$ by the maximal bound [F2], and ${\int }_0^t{G}^j{1}_{(0,\sigma ]}dB\to {\int }_0^{t}G{1}_{(0,\sigma ]}dB$ by the isometry and $\mathrm{\mid }{G}^j{1}_{(0,\sigma ]}-G{1}_{(0,\sigma ]}\mathrm{\mid }\le \mathrm{\mid }{G}^j-G\mathrm{\mid }$; passing to the limit in the identities of step 4.1 gives clause 4 in general, and since both sides are continuous in $t$ and agree at every deterministic $t$ almost surely, they are indistinguishable.

Agreement of stopped finite-energy integrals (clause 1, first assertion): for $m\ge n$ apply clause 4 to $G:=H{1}_{(0,{\tau }_m]}$, which has finite energy $E{A}_{t\wedge {\tau }_m}\le m$ by [F1], and to the stopping time ${\tau }_n$: $(M^{(m)}{)}_t^{{\tau }_n}=(G\cdot B{)}_{t\wedge {\tau }_n}={\int }_0^{t}G{1}_{(0,{\tau }_n]}dB={\int }_0^{t}H{1}_{(0,{\tau }_n]}dB=M_t^{(n)}$ almost surely for every $t$, using $1_{[0,{\tau }_n]}{1}_{(0,{\tau }_m]}=1_{(0,{\tau }_n]}$ because ${\tau }_n\le {\tau }_m$. Both sides are continuous, so $(M^{(m)}{)}^{{\tau }_n}$ and $M^{(n)}$ are indistinguishable.

Construction of $M$: put $M_t:={\lim }_n{M}_t^{(n)}$ where the limit exists finitely, and zero otherwise. This is progressive by [F6] applied on every finite-horizon product sigma-algebra, since each $M^{(n)}$ was chosen progressive in step 1.1. In particular $M_0=0$ everywhere and its stopped values are adapted by step 1.1. On the event $A$ where all the agreements of step 6.1 hold, every $M^{(n)}$ is continuous and ${\tau }_n\uparrow \mathrm{\infty }$, fix $\omega$ and $t$; choose $n$ with ${\tau }_n(\omega )\ge t$; then for every $m\ge n$, $M_t^{(m)}=M_{t\wedge {\tau }_n}^{(m)}=M_t^{(n)}$ by step 6.1, so the sequence is eventually constant and $M_t(\omega )=M_t^{(n)}(\omega )$. Hence on $A$ the process $M$ agrees on $[0,{\tau }_n(\omega )]$ with the continuous path of $M^{(n)}$, so $M$ has continuous paths on the full-measure event $A$.

$M$ is a local martingale with localizing sequence $({\tau }_n)$: by step 6.1 and the definition of $M$ on $A$, $M^{{\tau }_n}$ is indistinguishable from $M^{(n)}$, and $M^{(n)}$ is a martingale. Since $M^{{\tau }_n}$ is adapted by step 1.1 and has the same deterministic-time values almost surely, it is itself a martingale; moreover $E{M}_{t\wedge {\tau }_n}^2=E(M_t^{(n)}{)}^2=E{A}_{t\wedge {\tau }_n}\le n$ by [F1].

Clauses 2 and 3: if $N$ is continuous with $N_0=0$ and $N^{{\tau }_n}=M^{(n)}$ for all $n$, then for each $t$ and each $n$ with ${\tau }_n\ge t$ one has $N_t=N_{t\wedge {\tau }_n}=M_t^{(n)}=M_t$ almost surely, and letting $n\to \mathrm{\infty }$ along the full-measure event where ${\tau }_n\uparrow \mathrm{\infty }$ gives $N_t=M_t$ almost surely for every $t$; continuity and the rationals argument make $N$ indistinguishable from $M$. For clause 3, apply clause 4 twice: for each $k,n$, $(N^{{\rho }_k}{)}^{{\tau }_n}=(G_k\cdot B{)}^{{\tau }_n}={\int }_0^t{G}_k{1}_{(0,{\tau }_n]}dB$ and $(M^{(n)}{)}^{{\rho }_k}=(G_n\cdot B{)}^{{\rho }_k}={\int }_0^t{G}_n{1}_{(0,{\rho }_k]}dB$ with $G_k=H{1}_{(0,{\rho }_k]}$, $G_n=H{1}_{(0,{\tau }_n]}$, and both integrands equal $H{1}_{(0,{\rho }_k\wedge {\tau }_n]}$; hence $N^{({\rho }_k\wedge {\tau }_n)}$ and $M^{({\rho }_k\wedge {\tau }_n)}$ are indistinguishable, and for each $t$ on the full-measure event where ${\rho }_k\wedge {\tau }_n\ge t$ eventually, $N_t=M_t$ almost surely; continuity gives indistinguishability. The countable intersections of full events give the simultaneous identities; AC supplies the choices of versions and approximations through [F7].