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Cadlag Brownian-filtration local martingales have continuous versions

Statement

Assume the Axiom of Choice. Let $B$ be a standard Brownian motion with usual augmented natural filtration $(F_{t})$ Natural and usual augmented Brownian filtrations. Every local martingale $M$ relative to $(F_{t})$ whose paths are right-continuous with left limits on one event of probability one has a version with continuous paths, and any two continuous versions of $M$ are indistinguishable.

Facts & Assumptions

Given: AC, a standard Brownian motion $B$ with usual augmented filtration $(F_{t})$ , and a local martingale $M$ with cadlag paths.

[F1]

Representation. There is a predictable locally square-integrable $H$ with $M_{t} = M_{0} + \int_{0}^{t} H_{s} d B_{s}$ for all $t \geq 0$ up to indistinguishability, and such an $H$ is unique modulo $(d t \otimes P)$ -null sets on each finite horizon. Brownian-filtration martingale representation

[F2]

Continuity of localized integrals. For a predictable locally square-integrable $H$ the localized integral $t \mapsto \int_{0}^{t} H_{s} d B_{s}$ has continuous paths on a full-measure event and is unique up to indistinguishability among continuous processes with the same stopped finite-energy pieces. Localized Ito integral The Ito integral process has a continuous martingale version Locally square-integrable predictable Brownian integrands

[F3]

Indistinguishability from rational agreement. If two processes with continuous paths agree at every rational time on a single event of probability one, then they are indistinguishable: continuity extends the agreement to all times on that event. Process law, modification, and indistinguishability Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point

[F4]

AC bookkeeping. Choice is declared for the conditional-expectation interface underlying the representation. The Axiom of Choice

Proof

technique · direct

1.1

By [F1] write $M = M_{0} + \int H d B$ up to indistinguishability with $H$ predictable and locally square-integrable; by [F2] the localized integral has continuous paths on a full-measure event, so the process $M_{t}^{c} : = M_{0} + \int_{0}^{t} H_{s} d B_{s}$ is a continuous version of $M$ .

F1F2

2.1

Uniqueness: if $M^{'}$ and $M^{''}$ are two continuous versions of $M$ , then they agree with $M$ at every rational time almost surely, hence agree with each other at every rational time on the intersection of two full-measure events; by [F3] they are indistinguishable.

F3step 1.1

3.1

Boundary and consistency cases: for $M$ itself already continuous, the version is $M$ up to indistinguishability; for $M$ constant the integral representation has $H = 0$ ; the corollary shows that a cadlag local martingale of this filtration cannot have a genuine jump, because the representation is continuous; the uniqueness statement is about continuous versions, and no claim is made that an arbitrary cadlag modification is continuous pathwise; and AC enters only through [F4].

F1F2F4step 2.1∎

Source notes

Van der Vaart, Theorem 6.6, yields the continuity statement as an immediate consequence of the representation by a localized stochastic integral; the uniqueness argument is the standard rationals-and-continuity computation recorded in the definition of indistinguishability.

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Sources

Aad van der Vaart, Martingales, Diffusions and Financial Mathematics (preliminary notes), Theorem 6.6 (standard reference, not scraped)