Brownian reconstruction with a predictable elliptic coefficient

First work in the completed right-continuous augmentation of the given filtration. This preserves every continuous local martingale used here. Indeed, each original stopped true martingale $Q$ is uniformly integrable on $[0,T]$, since $Q_s=E[Q_T\mid {\mathcal{F}}_s]$; the uniform integrability follows by truncating the fixed integrable terminal variable. If $s_k\downarrow s$, continuity and uniform integrability give $Q_{s_k}\to Q_s$ in $L^1$. For an event in the completed intersection of the ${\mathcal{F}}_{s_k}$, its martingale event identity with $Q_t$ therefore passes to $Q_s$; completion adds only null-event identities. This proves the martingale property in the augmented filtration, while original localizers remain stopping times there. Countably many continuous processes can now be normalized on their common null exceptional set in the initial sigma-algebra. Let ${\tau }_n$ be the first time $\mathrm{\mid }M\mathrm{\mid }$ or $A$ reaches $n$, capped at $T$. These are increasing stopping times equal to $T$ eventually on each path in the common full event of continuity and finite energy, with $M^{{\tau }_n}$ bounded and $A_{T\wedge {\tau }_n}\le n$; the latter processes are continuous, and their hitting times are stopping times by progressive measurability and the rational-time test for continuous paths. By the bounded-stopping result in [F2], the original localizers show that $M^{{\tau }_n}$ is a bounded local martingale. Dominated convergence in its localizing event-integral identities, including at terminal time $T$, makes it a true square-integrable martingale. Work first with one such stopped martingale $N=M^{{\tau }_n}$, bracket density $c_s=1_{s\le {\tau }_n}{a}_s$, and the finite measure $\mu (E)=E{\int }_0^T{1}_E(s,\omega )\mathrm{tr}(c_s)ds$ restricted to the original predictable sigma-algebra. All elementary approximants below are predictable for that original filtration; the stopped integrator $N$ is used in the augmented space.

Bounded elementary predictable functions are dense in $L^2(\mu )$. To see this without replacing $\mu$ by $dt\otimes P$, let $V$ be their $L^2(\mu )$ closure. Each interval-generator indicator belongs to $V$, and time-zero indicators have zero $\mu$ norm. The constant class is represented by $1_{(0,T]}$. The sets whose indicators belong to $V$ form a lambda-system: complements follow by subtraction from this constant class; for disjoint unions, finite partial sums converge in $L^2(\mu )$ because the squared tail norm is the tail measure and $\mu$ is finite. Intersections of generators are generators or empty, so [F3] makes every predictable-set indicator belong to $V$. Simple approximation, truncation and dominated convergence then give all of $L^2(\mu )$. Apply this entrywise for matrices.

For a bounded elementary matrix $G={\sum }_k{G}_k{1}_{(t_k,t_{k+1}]}$ with $G_k$ measurable at $t_k$, define $I_N(G{)}_t={\sum }_k{G}_k(N_{t\wedge t_{k+1}}-N_{t\wedge t_k})$. Conditional martingale-increment identities and [F2] give $E\mathrm{\mid }{I}_N(G{)}_T{\mathrm{\mid }}^2=E{\int }_0^T\mathrm{tr}(G_s{c}_s{G}_s^{\mathsf{T}})ds,$ and $I_N(G{)}^i{I}_N(G{)}^j-{\int }_0^{\cdot }(G_s{c}_s{G}_s^{\mathsf{T}}{)}_{ij}ds$ is a martingale. The coordinate product-compensators of $N$ are true martingales: $N$ is bounded, and its bracket entries are bounded in absolute value by $A_{T\wedge {\tau }_n}\le n$, so the local martingales from [F2] are bounded and the preceding dominated-convergence argument applies. The displayed identities now follow on each block by multiplying those true martingale identities by bounded ${\mathcal{F}}_{t_k}$-measurable coefficients, and then summing; cross-block increments have conditional mean zero. For elementary $G$, the integral is bounded by a deterministic multiple of the bound on $N$, and its compensator is bounded by a deterministic multiple of $n$, so these product-compensators are true martingales as well. For two elementary matrices apply the same isometry to their difference and [F4] to obtain the maximal bound with factor $4$. Since $c$ is positive semidefinite, its energy is bounded by $E\int \mathrm{\parallel }G{\mathrm{\parallel }}_F^2\mathrm{tr}(c)ds=\mathrm{\parallel }G{\mathrm{\parallel }}_{L^2(\mu )}^2$.

Thus for every predictable $J\in L^2(\mu )$, elementary approximants from step 1.2 have integrals converging in $L^2$ of the path supremum to $I_N(J)$. A subsequence with summable expected sup errors converges uniformly almost surely to a continuous version. Define its value at each time by the pointwise limit of this subsequence when finite, and zero otherwise; this is adapted and progressive and has continuous paths on the common full convergence event, without modifying a process on a potentially noninitial null event. Martingale event identities pass to the $L^1$ limit. Products converge in $L^1$ by Cauchy–Schwarz; the compensators converge in expected total variation by the same inequality in the energy inner product, since $\mathrm{\parallel }J-G_m{\mathrm{\parallel }}_c\to 0$ and $\mathrm{\parallel }{G}_m{\mathrm{\parallel }}_c$ is bounded. Hence $[I_N(J){]}_t={\int }_0^t{J}_s{c}_s{J}_s^{\mathsf{T}}ds\mathrm{.}$ Indeed the right side is continuous finite variation and the product minus it is a martingale, so uniqueness follows from [F2]. Uniqueness also follows in the energy completion, and the construction extends to every predictable $J$ of finite $c$ energy by truncation: $J{1}_{\{\mathrm{\parallel }J\mathrm{\parallel }\le k\}}$ lies in $L^2(\mu )$ and converges in energy by dominated convergence.

Since $H$ is bounded and predictable, step 2.1 applies. Set $B^{(n)}=I_{M^{{\tau }_n}}(H)$. The constructions agree up to the smaller localizer. For elementary integrands the stopping identity is the defining increment identity, with the stopped integrator; extension follows by the maximal bound, since the energy restricted to a stopped interval is no greater than the full energy. This comparison also applies when the two approximation sequences differ, by uniqueness of the energy completion. Therefore the $B^{(n)}$ paste to an adapted continuous local martingale $B=\int HdM$, with $[B{]}_t={\int }_0^t{H}_s{a}_s{H}_s^{\mathsf{T}}ds=t{I}_d\mathrm{.}$ This first constructs $B$ in the augmented filtration; [A1] makes it Brownian there. It has an adapted version in the original filtration: choose an increasing subsequence of localizers, still denoted by ${\tau }_n$, with $P({\tau }_n<T)\le 2^{-n}$, and original elementary $G_n$ approximating $H$ in the corresponding stopped energy so well that $P({\sup }_{t\le T}\mathrm{\mid }{I}_{M^{{\tau }_n}}(G_n{)}_t-B_t^{{\tau }_n}\mathrm{\mid }>2^{-n})\le 2^{-n}$. Steps 1.2–2.1 provide these choices. The unstopped elementary sums $I_M(G_n)$ are adapted to the original filtration; on $\{{\tau }_n=T\}$ they equal the stopped sums. The union bound and summability therefore give uniform almost-sure convergence to $B$ on a common full event. Define the value at each time as their pointwise limit when finite and zero otherwise. This gives the desired original-adapted version, with the same continuous paths and bracket almost surely. Its Brownian increments in the augmented filtration are independent of every earlier event in the original subfiltration as well. Thus the original-adapted version is Brownian in the original filtration. The Lévy supplier's proof uses only the stated finite set of times in $[0,T]$, so it needs no extension past $T$.

We prove the recovery identity rather than assume an associativity rule. Fix $n$ and write $N=M^{{\tau }_n}$ and $Z=B^{{\tau }_n}=I_N(H)$. For bounded elementary $K$, its defining sum against $Z$ is $I_N(KH)$. On one block, this follows by approximating $H$ in step 2.1: multiplication by the bounded left-measurable block coefficient and restriction to that block commute with both elementary sums and their $L^2$ limits; sum over the finitely many blocks. The right integral exists because $KH$ is bounded predictable. For the stopped Brownian motion $Z$, the coordinate product-compensators are also true martingales. Brownian Doob bounds give $E{\sup }_{t\le T}\mathrm{\mid }{B}_t{\mathrm{\mid }}^2<\mathrm{\infty }$, so stopping the Brownian coordinate and product martingales is justified by the same finite-valued approximation and $L^1$ domination. Therefore the elementary isometry and maximal bound of step 1.3 apply to $Z$ as well. Since $[Z{]}_t=(t\wedge {\tau }_n)I_d$, elementary integrals against $Z$ have energy $E{\int }_0^{T\wedge {\tau }_n}\mathrm{\parallel }{K}_s{\mathrm{\parallel }}_F^{2}ds$. Applying steps 1.2–2.1 with this finite measure defines their completion, namely the stopped predictable Brownian integral. Choose elementary $K_j$ converging to $\sigma$ for that energy; this is possible since $E{\int }_0^{T\wedge {\tau }_n}\mathrm{\parallel }{\sigma }_s{\mathrm{\parallel }}_F^{2}ds\le n$. For the right integrals the exact identity $E{\int }_0^{T\wedge {\tau }_n}\mathrm{tr}((K_j-\sigma )Ha{H}^{\mathsf{T}}(K_j-\sigma {)}^{\mathsf{T}})ds=E{\int }_0^{T\wedge {\tau }_n}\mathrm{\parallel }{K}_j-\sigma {\mathrm{\parallel }}_F^{2}ds$ shows convergence to $I_N(\sigma H)=I_N(I_d)=N$. The left sums converge to $\int \sigma dZ$ in the same maximal norm. Thus ${\int }_0^{t\wedge {\tau }_n}{\sigma }_{s}d{B}_s=M_{t\wedge {\tau }_n}$. Letting $n$ increase proves the recovery identity on all of $[0,T]$, with the usual predictable Brownian integral defined by these localized elementary completions.