Null-Set Completion and Right-Continuous Brownian Filtrations

Three Filtrations, Two Different Properties

Brownian martingale representation is often first established for the filtration generated by Brownian motion. In applications, however, filtrations are usually completed by adding subsets of probability-zero events and made right-continuous. The representation property survives these usual augmentations. The reason is that completion only changes random variables up to almost-sure equality, while right-continuous augmentation adds no new information to the natural Brownian filtration.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). Let \(\mathcal{F}^0_t=\sigma(B_s:0\le s\le t)\) be its raw natural filtration. Let \(\mathcal{N}\) denote all subsets of \(\mathbb{P}\)-null sets in \(\mathcal{F}\), and define the completed natural filtration by \(\overline{\mathcal{F}}_t=\sigma(\mathcal{F}^0_t\cup\mathcal{N})\). Its right-continuous augmentation is \(\mathcal{F}_t=\bigcap_{u>t}\overline{\mathcal{F}}_u\). Equalities of sigma-fields below are understood up to completion, or equivalently up to events that differ by a null set.

Why the Completed Brownian Filtration Is Right-Continuous

The key Brownian fact is that observing the path immediately after time \(t\) adds no extra information at time \(t\), once null sets have been included. Here is a proof of the fact in the form needed for representation. We use the standard density of bounded continuous functions of finitely many Brownian coordinates in \(L^2\) of the Brownian path sigma-field; it follows by approximating measurable functions of finite coordinate vectors and then using that those coordinates generate the path sigma-field.

Proof. Choose \(u_n=t+1/n\), and write \(\mathcal{G}_n=\overline{\mathcal{F}}_{u_n}\). These sigma-fields decrease. Their intersection is \(\bigcap_{u>t}\overline{\mathcal{F}}_u\): for any \(u>t\), some \(u_n<u\), so \(\mathcal{G}_n\subseteq\overline{\mathcal{F}}_u\); the reverse inclusion follows because every \(u_n>t\).

For \(X\in L^2(\overline{\mathcal{F}}_T)\), the reverse martingale convergence theorem gives \(\mathbb{E}[X\mid\mathcal{G}_n]\to\mathbb{E}[X\mid\bigcap_n\mathcal{G}_n]\) in \(L^2\). We identify the same limit first for a bounded continuous cylinder variable \(X=g(B_{s_1},\ldots,B_{s_m})\). For all sufficiently large \(n\), the coordinates with \(s_i\le t\) are already observed at \(u_n\), and the remaining coordinates are at times \(s_i>u_n\). Conditional on \(\overline{\mathcal{F}}_{u_n}\), their future increments have the Brownian Gaussian distribution, independent of that sigma-field. Thus the conditional expectation is obtained by integrating \(g\) against the joint Gaussian law of those future increments, with starting value \(B_{u_n}\). As \(n\to\infty\), continuity of Brownian paths gives \(B_{u_n}\to B_t\), and the Gaussian increment means and covariances converge to those for increments starting at \(t\). Boundedness and continuity of \(g\) then give convergence in \(L^2\) to \(\mathbb{E}[X\mid\overline{\mathcal{F}}_t]\). If a coordinate time equals \(t\), its value is already known and the same argument applies.

Bounded continuous cylinder variables are dense in \(L^2(\overline{\mathcal{F}}_T)\), and conditional expectation is an \(L^2\) contraction. Therefore the convergence to \(\mathbb{E}[X\mid\overline{\mathcal{F}}_t]\) holds for every such \(X\). Comparing the two \(L^2\) limits shows that conditional expectation onto \(\bigcap_n\mathcal{G}_n\) equals conditional expectation onto \(\overline{\mathcal{F}}_t\) for every \(X\in L^2(\overline{\mathcal{F}}_T)\). In particular, applying this to indicators of events in the intersection shows that each such event agrees almost surely with an event in \(\overline{\mathcal{F}}_t\). The reverse inclusion is immediate. This proves equality up to completion, as asserted. \(\square\)

Representation Survives the Usual Augmentation

We use the \(L^2\) Brownian martingale representation theorem for the natural Brownian filtration: a square-integrable terminal variable measurable with respect to the Brownian path up to \(T\) has a representation by an Ito integral with a square-integrable predictable integrand. The result is the standard Brownian representation theorem established in the earlier material on martingale representation. The lemma shows that the right-continuous augmentation does not enlarge the completed natural Brownian filtration beyond its value at each time.

Proof. Let \(X\) be square-integrable and measurable with respect to the completed terminal sigma-field. Completion means there is an \(\mathcal{F}^0_T\)-measurable \(X_0\) with \(X=X_0\) almost surely. Apply the \(L^2\) representation theorem in the natural Brownian filtration to \(X_0\). It gives a predictable \(H\) with \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\) and \(X_0=\mathbb{E}[X_0]+\int_0^T H_s\,dB_s\). The same integrand is predictable in the augmented filtration. Its Ito integral there agrees with the original integral: both are defined by \(L^2\) limits of integrals of elementary predictable processes, and completion changes processes only up to the relevant almost-sure identifications. Since \(X=X_0\) almost surely, the representation holds for \(X\) as well.

Now let \(M_t\) be any square-integrable martingale adapted to the augmented filtration, and set \(X=M_T\). The martingale property gives \(M_t=\mathbb{E}[X\mid\mathcal{F}_t]\). The representation of \(X\) implies \(M_t=\mathbb{E}[X]+\int_0^t H_s\,dB_s\). To check the martingale step, the Ito integral is square-integrable by the Ito isometry, since its second moment is \(\mathbb{E}\int_0^t H_s^2\,ds<\infty\); its increments have conditional mean zero, first for elementary predictable integrands by Brownian independent increments and then for all such \(H\) by \(L^2\) approximation. Hence its conditional expectation at time \(t\) is its value at time \(t\). Also \(\mathbb{E}[X]=M_0\) almost surely because the augmented Brownian \(\mathcal{F}_0\) is trivial up to null sets. This proves the asserted representation. \(\square\)

Completeness Does Not Guarantee Representation

Completion makes null events measurable; it does not prevent a filtration from containing additional information that is not generated by \(B_t\). A filtration can be complete and still fail to have the predictable representation property with respect to a given Brownian motion. The following example has trivial initial sigma-field, so the failure is not merely random information already present at time zero.

What the Augmentation Does—and Does Not Do

For the natural Brownian filtration, completion permits arbitrary almost-sure modifications of terminal variables without changing their representing integrands, while right-continuity adds no new information, by the lemma. Consequently the predictable representation property passes to the usual augmented filtration. A martingale in this statement is a true square-integrable martingale, not merely a local martingale: square-integrability supplies the terminal variable and the integrability needed for the \(L^2\) representation theorem. Local martingale representation is a separate localization statement.

Keep the two notions separate. Probability-space completeness concerns subsets of null events. The predictable representation property concerns whether every martingale in a filtration can be expressed using stochastic integrals against the specified Brownian motion (and its initial value). Adding information not generated by that Brownian motion can destroy the property even when the space is complete. Neither completion nor right-continuity, by itself, is a substitute for checking what information the filtration contains.