LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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Adapted continuous processes are progressively measurable

Statement

Let $(Ω, F, P)$ be a probability space with a continuous-time filtration $(F_{t})_{t \geq 0}$ , and let $X = (X_{t})_{t \geq 0}$ be a real process that satisfies $X_{t} : Ω \to R$ is $F_{t}$ -measurable for every $t \geq 0$ , and has continuous paths, in the strong sense that $s \mapsto X_{s} (ω)$ is continuous on $[0, \infty)$ for every $ω \in Ω$ . Then $X$ is progressively measurable and predictable relative to $(F_{t})$ in the sense of Progressively measurable and predictable processes, including at time zero under the generator convention ${0} \times A$ , $A \in F_{0}$ . No choice principle is used. Here a filtration means an increasing family of sub-sigma-algebras of $F$ . The fixed-time measurability hypothesis is the adaptedness terminology of clause 1 of Continuous-time adapted processes and martingales; only that clause is used, not its conditional-expectation or martingale interface and its Choice assumption.

If instead the paths are continuous only on an event $A \in F$ with $P (A) = 1$ , the conclusion holds for the modification that is set equal to $0$ off $A$ provided $A \in F_{0}$ ; without such a measurability assumption on the continuity event no predictability claim is made, because predictability is a property of the given joint map.

Facts & Assumptions

Given: a probability space with a filtration $(F_{t})_{t \geq 0}$ , a real adapted process $X$ with continuous paths everywhere, a horizon $T > 0$ , and for $n \geq 1$ the dyadic grid $t_{k} = k T / 2^{n}$ , $0 \leq k \leq 2^{n}$ .

[F1]

$X$ is adapted: $X_{u}$ is $F_{u}$ -measurable for every $u \geq 0$ ; in particular $X_{u}$ is $F_{T}$ -measurable for $u \leq T$ . Continuous-time adapted processes and martingales

[F2]

Each rectangle $C \times A$ , where $C \in B ([0, T])$ and $A \in F_{u}$ for some $u \leq T$ , belongs to $B ([0, T]) \otimes F_{T}$ , since $F_{u} \subseteq F_{T}$ . Finite unions of these rectangles also belong to that sigma-algebra. Progressively measurable and predictable processes

[F3]

A pointwise limit of measurable functions into $R$ is measurable; the same theorem applied coordinatewise gives measurability of limits of jointly measurable maps. Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable

[F4]

The generators of the predictable sigma-algebra are the sets $(s, u] \times A$ with $A \in F_{s}$ and the time-zero sets ${0} \times A$ with $A \in F_{0}$ . The set $[0, T] \times Ω$ is predictable, being the union of the generator $(0, T] \times Ω$ and the time-zero generator ${0} \times Ω$ . Progressively measurable and predictable processes

Proof

technique · direct

1.1

Fix $n \geq 1$ and define, for $(s, ω) \in [0, T] \times Ω$ , $H_{s}^{n} (ω) : = \sum_{k = 0}^{2^{n} - 1} X_{t_{k}} (ω) 1_{(t_{k}, t_{k + 1}]} (s) + X_{0} (ω) 1_{{0}} (s) .$ For a Borel set $B \subseteq R$ the preimage is ${H^{n} \in B} = ({0} \times {X_{0} \in B}) \cup ⋃_{k = 0}^{2^{n} - 1} ((t_{k}, t_{k + 1}] \times {X_{t_{k}} \in B})$ , a finite union of measurable rectangles of $B ([0, T]) \otimes F_{T}$ , since $X_{t_{k}}$ is $F_{t_{k}}$ -measurable and $F_{t_{k}} \subseteq F_{T}$ .

F1F2given

2.1

Each $H^{n}$ is measurable for $B ([0, T]) \otimes F_{T}$ , and for every $(s, ω)$ the identity $H_{s}^{n} (ω) \to X_{s} (ω)$ holds: at $s = 0$ both sides equal $X_{0} (ω)$ , and for $s \in (0, T]$ the left endpoints $t_{k}$ of the dyadic intervals containing $s$ tend to $s$ , so path continuity gives $X_{t_{k}} (ω) \to X_{s} (ω)$ .

step 1.1given

2.2

Each $H^{n}$ is predictable: the preimage formula of step 1.1 exhibits each Borel inverse image as a finite union of generators of the predictable sigma-algebra, since $X_{t_{k}}$ is $F_{t_{k}}$ -measurable and each interval $(t_{k}, t_{k + 1}]$ is a generator interval, while the time-zero term is ${0} \times {X_{0} \in B}$ with ${X_{0} \in B} \in F_{0}$ .

F1F4step 1.1

3.1

By [F3] the pointwise limit $X$ restricted to $[0, T] \times Ω$ is $B ([0, T]) \otimes F_{T}$ -measurable; $T > 0$ was arbitrary, so $X$ is progressively measurable.

F3step 2.1

3.2

For each fixed $T$ the restriction of $X$ to $[0, T] \times Ω$ is a pointwise limit of the predictable processes $H^{n}$ restricted to $[0, T]$ , hence is predictable on that horizon by [F3]; and the horizon- $T$ pieces assemble to a globally predictable process because $[0, T] \times Ω \in P$ and a set is predictable as soon as all its intersections with the countably many sets $[0, m] \times Ω$ , $m \geq 1$ , are.

F3F4step 2.2

4.1

Collecting steps 3.1 and 3.2, an adapted process with everywhere continuous paths is progressively measurable and predictable. The time-zero case is included: $H_{0}^{n} = X_{0}$ at every stage and the generator ${0} \times A$ , $A \in F_{0}$ , was used in step 2.2. For the final statement about $A \in F_{0}$ , the map $Y_{t} = X_{t} 1_{A}$ is $F_{t}$ -measurable (its Borel inverse images are $(A \cap {X_{t} \in B}) \cup (A^{c} if 0 \in B)$ ), has continuous paths everywhere, and agrees with $X$ on $A$ . Apply the conclusion to $Y$ . No grid point, approximant or limit in the argument is chosen: the dyadic grids and the left endpoints are fixed functions of $n$ , and pointwise limits are unique.

step 3.1step 3.2given∎

Source notes

The approximation is the standard dyadic-step argument of van der Vaart, Section 5.1; continuous adapted processes generate the predictable sigma-algebra, and the left-continuous staircase approximants are predictable by construction. The time-zero section uses exactly the generators ${0} \times A$ , $A \in F_{0}$ . The proof explicitly supplies the fixed-time measurable maps and does not invoke any conditional-expectation existence theorem.

Depends on

Used by

Logarithm of geometric Brownian motion Example
Brownian-filtration martingale representation Theorem
Integration by parts for Brownian Ito processes Theorem
Multidimensional Ito formula for Brownian-driven processes Theorem
One-dimensional Ito formula Theorem
Space-time harmonic functions yield Brownian local martingales up to exit lifetime Theorem

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Sources

Aad van der Vaart, Stochastic Integration and Differential Equations, Section 5.1 (standard reference, not scraped)